Research Article Chebyshev Type Integral Inequalities...

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Research Article Chebyshev Type Integral Inequalities Involving the Fractional Hypergeometric Operators D. Baleanu 1,2,3 and S. D. Purohit 4 1 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia 2 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey 3 Institute of Space Sciences, Magurele, Bucharest, Romania 4 Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur 313001, India Correspondence should be addressed to S. D. Purohit; sunil a [email protected] Received 31 January 2014; Accepted 8 March 2014; Published 22 April 2014 Academic Editor: Juan J. Nieto Copyright © 2014 D. Baleanu and S. D. Purohit. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By making use of the fractional hypergeometric operators, we establish certain new fractional integral inequalities for synchronous functions which are related to the weighted version of the Chebyshev functional. Some consequent results and special cases of the main results are also pointed out. 1. Introduction Fractional integral inequalities have proved to be one of the most important and powerful tools for the development of many branches of pure and applied mathematics. ese inequalities have many applications in numerical quadrature, transform theory, probability, and statistical problems, but the most useful ones are in establishing uniqueness of solutions in fractional boundary value problems. Moreover, they also provide upper and lower bounds to the solutions of the above equations. erefore, a significant development in the classical and fractional integral inequalities, particularly in analysis, has been witnessed; see, for instance, the papers [1–7] and the references cited therein. Let the functional (, ) = 1 − ∫ () () −( 1 − ∫ () )( 1 − ∫ () ), (1) where and are two integrable functions and synchronous on [, ]; that is, ( () − ()) ( () − ()) ≥ 0, (2) for any , ∈ [, ]; then the Chebyshev integral inequality [8] is given by (,) ≥ 0. e sign of this inequality is reversed if and are asynchronous on [, ] (i.e., (() − ())(() − ()) ≤ 0, for any , ∈ [, ]). Under various assumptions (Chebyshev inequalities, Gr¨ uss inequality, etc.) [9–11], Chebyshev functionals are useful to give a lower bound or an upper bound for (, ), in the theory of approximations. erefore, in the literature, we found several papers that analyze extensions and generalizations of these integral inequalities by involving fractional calculus and - calculus operators. One may refer to such type of works in the book [12] and the recent papers [13–24]. A similar type of important role is also played by the Chebyshev polynomials of first and second kind in the theory of approximation. For a detailed account about these polynomials and their generating functions, one can see a recent paper [25] and research monograph by Srivastava and Manocha [26]. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 609160, 10 pages http://dx.doi.org/10.1155/2014/609160

Transcript of Research Article Chebyshev Type Integral Inequalities...

Page 1: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

Research ArticleChebyshev Type Integral Inequalities Involvingthe Fractional Hypergeometric Operators

D Baleanu123 and S D Purohit4

1 Department of Chemical and Materials Engineering Faculty of Engineering King Abdulaziz University PO Box 80204Jeddah 21589 Saudi Arabia

2Department of Mathematics and Computer Sciences Faculty of Arts and Sciences Cankaya University 06530 Ankara Turkey3 Institute of Space Sciences Magurele Bucharest Romania4Department of Basic Sciences (Mathematics) College of Technology and EngineeringMP University of Agriculture and Technology Udaipur 313001 India

Correspondence should be addressed to S D Purohit sunil a purohityahoocom

Received 31 January 2014 Accepted 8 March 2014 Published 22 April 2014

Academic Editor Juan J Nieto

Copyright copy 2014 D Baleanu and S D PurohitThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

By making use of the fractional hypergeometric operators we establish certain new fractional integral inequalities for synchronousfunctions which are related to the weighted version of the Chebyshev functional Some consequent results and special cases of themain results are also pointed out

1 Introduction

Fractional integral inequalities have proved to be one ofthe most important and powerful tools for the developmentof many branches of pure and applied mathematics Theseinequalities have many applications in numerical quadraturetransform theory probability and statistical problems butthe most useful ones are in establishing uniqueness ofsolutions in fractional boundary value problems Moreoverthey also provide upper and lower bounds to the solutions ofthe above equations Therefore a significant development inthe classical and fractional integral inequalities particularlyin analysis has been witnessed see for instance the papers[1ndash7] and the references cited therein

Let the functional

119879 (119891 119892) =1

119887 minus 119886int

119887

119886

119891 (119909) 119892 (119909) 119889119909

minus (1

119887 minus 119886int

119887

119886

119891 (119909) 119889119909)(1

119887 minus 119886int

119887

119886

119892 (119909) 119889119909)

(1)

where 119891 and 119892 are two integrable functions and synchronouson [119886 119887] that is

(119891 (119909) minus 119891 (119910)) (119892 (119909) minus 119892 (119910)) ge 0 (2)

for any 119909 119910 isin [119886 119887] then the Chebyshev integral inequality[8] is given by 119879(119891 119892) ge 0 The sign of this inequality isreversed if 119891 and 119892 are asynchronous on [119886 119887] (ie (119891(119909) minus119891(119910))(119892(119909) minus 119892(119910)) le 0 for any 119909 119910 isin [119886 119887]) Under variousassumptions (Chebyshev inequalities Gruss inequality etc)[9ndash11] Chebyshev functionals are useful to give a lowerbound or an upper bound for 119879(119891 119892) in the theory ofapproximationsTherefore in the literature we found severalpapers that analyze extensions and generalizations of theseintegral inequalities by involving fractional calculus and 119902-calculus operators One may refer to such type of works inthe book [12] and the recent papers [13ndash24] A similar type ofimportant role is also played by the Chebyshev polynomialsof first and second kind in the theory of approximationFor a detailed account about these polynomials and theirgenerating functions one can see a recent paper [25] andresearch monograph by Srivastava and Manocha [26]

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 609160 10 pageshttpdxdoiorg1011552014609160

2 Abstract and Applied Analysis

For the present paper let us consider theweighted versionof the Chebyshev functional (see [8])

119879 (119891 119892 119901) = int

119887

119886

119901 (119909) 119889119909int

119887

119886

119891 (119909) 119892 (119909) 119901 (119909) 119889119909

minus int

119887

119886

119891 (119909) 119901 (119909) 119889119909int

119887

119886

119892 (119909) 119901 (119909) 119889119909

(3)

provided that 119891 and 119892 are two integrable functions on [119886 119887]and 119901(119909) is a positive and integrable function on [119886 119887] In2000 Dragomir [27] derived the following inequality

21003816100381610038161003816119879 (119891 119892 119901)

1003816100381610038161003816 le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

[∬

119887

119886

1003816100381610038161003816119909 minus 1199101003816100381610038161003816 119901 (119909) 119901 (119910) 119889119909 119889119910]

(4)

where 119891 119892 are two differentiable functions and 1198911015840 isin 119871119903(119886 119887)1198921015840

isin 119871 119904(119886 119887) 119903 gt 1 119903minus1 + 119904minus1

= 1 RecentlyDahmani et al [28] established some integral results relatedto Chebyshevrsquos functional (3) in the case of differentiablefunctions whose derivatives belong to the space 119871119901([0infin))involvingRiemann-Liouville fractional integrals Purohit andRaina [22 29] added one more dimension to this study byintroducing certain new integral inequalities for synchronousfunctions involving the Saigo fractional integral operators[30] Further Baleanu et al [24] established certain general-ized integral inequalities for synchronous functions that arerelated to the Chebyshev functional (1) using the fractionalhypergeometric operator introduced by Curiel and Galue[31]

In this paper we establish certain integral inequalitiesrelated to the weighted Chebyshevrsquos functional (3) in the caseof differentiable functions whose derivatives belong to thespace 119871119901([0infin)) involving fractional hypergeometric oper-ators due to [31] Later we develop some integral inequalitiesfor the fractional integrals by suitably choosing the function119901(119905) Some of the known results due to Dahmani et al [28]and Purohit and Raina [29] follow as special cases of ourfindings

Firstly we give some necessary definitions and mathe-matical preliminaries of fractional calculus operators whichare used further in this paper

Definition 1 A real-valued function 119891(119905) (119905 gt 0) is said to bein the space C120583 (120583 isin R) if there exists a real number 119901 gt 120583

such that 119891(119905) = 119905119901120601(119905) where 120601(119905) isin C(0infin)

Definition 2 Let 120572 gt 0 120583 gt minus1 120573 120578 isin R thena generalized fractional integral 119868120572120573120578120583

119905(in terms of the

Gauss hypergeometric function) of order 120572 for a real-valuedcontinuous function 119891(119905) is defined by [31] (see also [32])

119868120572120573120578120583

119905119891 (119905) =

119905minus120572minus120573minus2120583

Γ (120572)

times int

119905

0

120591120583(119905 minus 120591)

120572minus1

times21198651 (120572+120573+120583 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(5)

where the function 21198651(minus) appearing as a kernel for theoperator (5) is the Gaussian hypergeometric function definedby

21198651 (119886 119887 119888 119905) =

infin

sum

119899=0

(119886)119899(119887)119899

(119888)119899

119905119899

119899 (6)

and (119886)119899 is the Pochhammer symbol

(119886)119899 = 119886 (119886 + 1) sdot sdot sdot (119886 + 119899 minus 1) (119886)0 = 1 (7)

It may be noted that the Pochhammer symbol in terms of thegamma function is defined by

(119886)119899 =Γ (119886 + 119899)

Γ (119886) (119899 gt 0) (8)

where the gamma function [33] is given by

Γ (119909) = int

infin

0

119905119909minus1

119890minus119905119889119905 (R (119909) gt 0) (9)

Our results in this paper are based on the following pre-liminary assertions giving composition formula of fractionalintegral (5) with a power function

Lemma 3 (see [24]) Let 120572 120573 120578 120582 isin R 120583 gt minus1 120583 + 120582 gt 0and 120582 minus 120573 + 120578 gt 0 then the following image formula for thepower function under the operator (5) holds true

119868120572120573120578120583

119905119905120582minus1

=Γ (120583 + 120582) Γ (120582 minus 120573 + 120578)

Γ (120582 minus 120573) Γ (120582 + 120583 + 120572 + 120578)119905120582minus120573minus120583minus1

(10)

2 Main Results

In this section we obtain certain integral inequality whichgives estimation for the fractional integral of a product interms of the product of the individual function fractionalintegrals involving fractional hypergeometric operators Wegive our results related to Chebyshevrsquos functional (3) in thecase of differentiable mappings whose derivatives belong tothe space 119871119901([0infin)) satisfying Holderrsquos inequality

Theorem 4 Let 119901 be a positive function and let 119891 and 119892 betwo synchronous functions on [0infin) If 1198911015840 isin 119871119903([0infin))

Abstract and Applied Analysis 3

1198921015840isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904

minus1= 1 then (for all 119905 gt 0

120573 lt 1 120583 gt minus1 120572 gt max0 minus120573 minus 120583 120573 minus 1 lt 120578 lt 0)

2100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(11)

Proof Let119891 and 119892 be two synchronous functions then usingDefinition 1 for all 120591 120588 isin (0 119905) 119905 ge 0 we define

H (120591 120588) = (119891 (120591) minus 119891 (120588)) (119892 (120591) minus 119892 (120588)) (12)

Consider

119865 (119905 120591) =119905minus120572minus120573minus2120583

120591120583(119905 minus 120591)

120572minus1

Γ (120572)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

(120591 isin (0 119905) 119905 gt 0)

=120591120583

Γ (120572)

(119905 minus 120591)120572minus1

119905120572+120573+2120583+120591120583(120572 + 120573 + 120583) (minus120578)

Γ (120572 + 1)

(119905 minus 120591)120572

119905120572+120573+2120583+1

+120591120583(120572 + 120573 + 120583) (120572 + 120573 + 120583 + 1) (minus120578) (minus120578 + 1)

2Γ (120572 + 2)

times(119905 minus 120591)

120572+1

119905120572+120573+2120583+2+ sdot sdot sdot

(13)

We observe that each term of the above series is positive inview of the conditions stated withTheorem 4 and hence thefunction 119865(119905 120591) remains positive for all 120591 isin (0 119905) (119905 gt 0)

Multiplying both sides of (12) by 119865(119905 120591)119901(120591) (where119865(119905 120591) is given by (13)) and integrating with respect to 120591 from0 to 119905 and using (5) we get

119905minus120572minus120573minus2120583

Γ (120572)

times int

119905

0

120591120583(119905 minus 120591)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905) 119901 (120591)H (120591 120588) 119889120591

= 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119891 (120588) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

minus 119892 (120588) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) + 119891 (120588) 119892 (120588) 119868

120572120573120578120583

119905119901 (119905)

(14)

Next on multiplying both sides of (14) by 119865(119905 120588)119901(120588) where119865(119905 120588) is given by (13) and integrating with respect to 120588 from0 to 119905 we can write

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 2 (119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905))

(15)

In view of (12) we have

H (120591 120588) = ∬

120588

120591

1198911015840(119910) 1198921015840(119911) 119889119910 119889119911 (16)

Using the followingHolder inequality for the double integral10038161003816100381610038161003816100381610038161003816

∬

120588

120591

119891 (119910) 119892 (119911) 119889119910 119889119911

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

1003816100381610038161003816119891(119910)1003816100381610038161003816

119903119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119904119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(119903minus1+ 119904minus1= 1)

(17)

we obtain

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(18)

4 Abstract and Applied Analysis

Since

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

=1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

=1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

119904minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(19)

then (18) reduces to

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 le

1003816100381610038161003816120591 minus 1205881003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(20)

It follows from (15) that

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591minus120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(21)

Applying againHolderrsquos inequality (17) on the right-hand sideof (21) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120572minus119904120573minus2119904120583

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(22)

In view of the fact that

10038161003816100381610038161003816100381610038161003816

int

120588

120591

1003816100381610038161003816119891(119910)1003816100381610038161003816

119901119889119910

10038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171198911003817100381710038171003817

119901

119901 (23)

we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le [

[

119905minus119903120572minus119903120573minus211990312058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817

119903

119903

Γ119903 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

Abstract and Applied Analysis 5

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119903minus1

times [

[

119905minus119904120572minus119904120573minus211990412058310038171003817100381710038171003817

119892101584010038171003817100381710038171003817

119904

119904

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119904minus1

(24)

From (24) we obtain

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119903minus1

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119904minus1

(25)

Since 119903minus1 + 119904minus1 = 1 the above inequality yields

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(26)

which in view of (15) gives

2100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

(27)

Making use of (26) and (27) the left-hand side of theinequality (11) follows

To prove the right-hand side of the inequality (11) weobserve that 0 le 120591 le 119905 0 le 120588 le 119905 and therefore

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (28)

6 Abstract and Applied Analysis

Evidently from (26) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588) 119889120591 119889120588

=10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(29)

which completes the proof of Theorem 4

The following gives a generalization of Theorem 4

Theorem 5 Let 119901 be a positive function and let 119891 and 119892 betwo synchronous functions on [0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 = 1 then

100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(30)

for all 119905 gt 0 120572 gt max0 minus120573minus120583 120573 lt 1 120583 gt minus1 120573minus1 lt 120578 lt 0120574 gt max0 minus120575 minus ] 120575 lt 1 ] gt minus1 120575 minus 1 lt 120577 lt 0

Proof To prove the above theorem we use inequality (14)Multiplying both sides of (14) by

119905minus120574minus120575minus2]

120588](119905 minus 120588)

120574minus1

Γ (120574)21198651 (120574 + 120575 + ] minus120577 120574 1 minus

120588

119905) 119901 (120588)

(120588 isin (0 119905) 119905 gt 0)

(31)

which remains positive in view of the conditions stated with(30) and then integrating with respect to 120588 from 0 to 119905 we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus 119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

(32)

Now making use of (20) then (32) gives

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

Abstract and Applied Analysis 7

le119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

times

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(33)

Applying Holderrsquos inequality (17) on the right-hand side of(33) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120574minus119904120575minus2119904]

Γ119904 (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + 120583 minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(34)

which on using (23) readily yields the following inequality

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(35)

In view of (32) and (35) and the properties of modulus onecan easily arrive at the left-sided inequality of Theorem 5Moreover we have 0 le 120591 le 119905 0 le 120588 le 119905 hence

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (36)

Therefore from (35) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

8 Abstract and Applied Analysis

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588) 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(37)

which completes the proof of Theorem 5

Remark 6 For 120574 = 120572 120575 = 120573 120577 = 120578 and ] = 120583 Theorem 5immediately reduces to Theorem 4

3 Consequent Results and Special Cases

As implications of our main results we consider someconsequent results of Theorems 4 and 5 by suitably choosingthe function 119901(119905) To this end let us set 119901(119905) = 119905

120582(120582 isin

[0infin) 119905 isin (0infin)) then on using (10) Theorems 4 and 5yield the following results

Corollary 7 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120582 + 120583 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583+120582

120588120583+120582

(119905 minus 120591)120572minus1

(119905 minus 120588)120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 120582 + 1) Γ

2(120582 + 1 minus 120573 + 120578) 119905

1+2120582minus2120573minus2120583

Γ2 (120582 + 1 minus 120573) Γ2 (120583 + 120582 + 1 + 120572 + 120578)

(38)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120582 ge 0min(120582 + 120583 120582 minus 120573 + 120578) gt minus1

Corollary 8 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120574120575120577]119905

119905120582119891 (119905) 119892 (119905)

+Γ (] + 120582 + 1) Γ (120582 + 1 minus 120575 + 120577)

Γ (120582 + 1 minus 120575) Γ (] + 120582 + 1 + 120574 + 120577)119905120582minus120575minus]

times 119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119905120582119891 (119905) 119868

120574120575120577]119905

119905120582119892 (119905)

minus119868120574120575120577]119905

119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574+120575+] minus120577 120574 1minus120588

119905) 120591120583+120582

120588]+120582 1003816100381610038161003816120591minus120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

times (Γ (120583 + 120582 + 1) Γ (] + 120582 + 1)

timesΓ (120582 + 1 minus 120573 + 120578) Γ (120582 + 1 minus 120575 + 120577))

times (Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

timesΓ(120582 + 1 minus 120575)Γ(] + 120582 + 1 + 120574 + 120577))minus1

times 1199051+2120582minus120573minus120575minus2120583minus2]

(39)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0 120582 ge 0min(120582 + 120583 120582 + ] 120582 minus 120573 + 120578 120582 minus 120575 + 120577) gt minus1

Further if we put 120582 = 0 in Corollaries 7 and 8 (or 119901(119905) =1 in Theorems 4 and 5) we obtain the following integralinequalities

Corollary 9 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)119905minus120573minus120583

119868120572120573120578120583

119905119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119891 (119905) 119868

120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

2 Abstract and Applied Analysis

For the present paper let us consider theweighted versionof the Chebyshev functional (see [8])

119879 (119891 119892 119901) = int

119887

119886

119901 (119909) 119889119909int

119887

119886

119891 (119909) 119892 (119909) 119901 (119909) 119889119909

minus int

119887

119886

119891 (119909) 119901 (119909) 119889119909int

119887

119886

119892 (119909) 119901 (119909) 119889119909

(3)

provided that 119891 and 119892 are two integrable functions on [119886 119887]and 119901(119909) is a positive and integrable function on [119886 119887] In2000 Dragomir [27] derived the following inequality

21003816100381610038161003816119879 (119891 119892 119901)

1003816100381610038161003816 le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

[∬

119887

119886

1003816100381610038161003816119909 minus 1199101003816100381610038161003816 119901 (119909) 119901 (119910) 119889119909 119889119910]

(4)

where 119891 119892 are two differentiable functions and 1198911015840 isin 119871119903(119886 119887)1198921015840

isin 119871 119904(119886 119887) 119903 gt 1 119903minus1 + 119904minus1

= 1 RecentlyDahmani et al [28] established some integral results relatedto Chebyshevrsquos functional (3) in the case of differentiablefunctions whose derivatives belong to the space 119871119901([0infin))involvingRiemann-Liouville fractional integrals Purohit andRaina [22 29] added one more dimension to this study byintroducing certain new integral inequalities for synchronousfunctions involving the Saigo fractional integral operators[30] Further Baleanu et al [24] established certain general-ized integral inequalities for synchronous functions that arerelated to the Chebyshev functional (1) using the fractionalhypergeometric operator introduced by Curiel and Galue[31]

In this paper we establish certain integral inequalitiesrelated to the weighted Chebyshevrsquos functional (3) in the caseof differentiable functions whose derivatives belong to thespace 119871119901([0infin)) involving fractional hypergeometric oper-ators due to [31] Later we develop some integral inequalitiesfor the fractional integrals by suitably choosing the function119901(119905) Some of the known results due to Dahmani et al [28]and Purohit and Raina [29] follow as special cases of ourfindings

Firstly we give some necessary definitions and mathe-matical preliminaries of fractional calculus operators whichare used further in this paper

Definition 1 A real-valued function 119891(119905) (119905 gt 0) is said to bein the space C120583 (120583 isin R) if there exists a real number 119901 gt 120583

such that 119891(119905) = 119905119901120601(119905) where 120601(119905) isin C(0infin)

Definition 2 Let 120572 gt 0 120583 gt minus1 120573 120578 isin R thena generalized fractional integral 119868120572120573120578120583

119905(in terms of the

Gauss hypergeometric function) of order 120572 for a real-valuedcontinuous function 119891(119905) is defined by [31] (see also [32])

119868120572120573120578120583

119905119891 (119905) =

119905minus120572minus120573minus2120583

Γ (120572)

times int

119905

0

120591120583(119905 minus 120591)

120572minus1

times21198651 (120572+120573+120583 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(5)

where the function 21198651(minus) appearing as a kernel for theoperator (5) is the Gaussian hypergeometric function definedby

21198651 (119886 119887 119888 119905) =

infin

sum

119899=0

(119886)119899(119887)119899

(119888)119899

119905119899

119899 (6)

and (119886)119899 is the Pochhammer symbol

(119886)119899 = 119886 (119886 + 1) sdot sdot sdot (119886 + 119899 minus 1) (119886)0 = 1 (7)

It may be noted that the Pochhammer symbol in terms of thegamma function is defined by

(119886)119899 =Γ (119886 + 119899)

Γ (119886) (119899 gt 0) (8)

where the gamma function [33] is given by

Γ (119909) = int

infin

0

119905119909minus1

119890minus119905119889119905 (R (119909) gt 0) (9)

Our results in this paper are based on the following pre-liminary assertions giving composition formula of fractionalintegral (5) with a power function

Lemma 3 (see [24]) Let 120572 120573 120578 120582 isin R 120583 gt minus1 120583 + 120582 gt 0and 120582 minus 120573 + 120578 gt 0 then the following image formula for thepower function under the operator (5) holds true

119868120572120573120578120583

119905119905120582minus1

=Γ (120583 + 120582) Γ (120582 minus 120573 + 120578)

Γ (120582 minus 120573) Γ (120582 + 120583 + 120572 + 120578)119905120582minus120573minus120583minus1

(10)

2 Main Results

In this section we obtain certain integral inequality whichgives estimation for the fractional integral of a product interms of the product of the individual function fractionalintegrals involving fractional hypergeometric operators Wegive our results related to Chebyshevrsquos functional (3) in thecase of differentiable mappings whose derivatives belong tothe space 119871119901([0infin)) satisfying Holderrsquos inequality

Theorem 4 Let 119901 be a positive function and let 119891 and 119892 betwo synchronous functions on [0infin) If 1198911015840 isin 119871119903([0infin))

Abstract and Applied Analysis 3

1198921015840isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904

minus1= 1 then (for all 119905 gt 0

120573 lt 1 120583 gt minus1 120572 gt max0 minus120573 minus 120583 120573 minus 1 lt 120578 lt 0)

2100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(11)

Proof Let119891 and 119892 be two synchronous functions then usingDefinition 1 for all 120591 120588 isin (0 119905) 119905 ge 0 we define

H (120591 120588) = (119891 (120591) minus 119891 (120588)) (119892 (120591) minus 119892 (120588)) (12)

Consider

119865 (119905 120591) =119905minus120572minus120573minus2120583

120591120583(119905 minus 120591)

120572minus1

Γ (120572)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

(120591 isin (0 119905) 119905 gt 0)

=120591120583

Γ (120572)

(119905 minus 120591)120572minus1

119905120572+120573+2120583+120591120583(120572 + 120573 + 120583) (minus120578)

Γ (120572 + 1)

(119905 minus 120591)120572

119905120572+120573+2120583+1

+120591120583(120572 + 120573 + 120583) (120572 + 120573 + 120583 + 1) (minus120578) (minus120578 + 1)

2Γ (120572 + 2)

times(119905 minus 120591)

120572+1

119905120572+120573+2120583+2+ sdot sdot sdot

(13)

We observe that each term of the above series is positive inview of the conditions stated withTheorem 4 and hence thefunction 119865(119905 120591) remains positive for all 120591 isin (0 119905) (119905 gt 0)

Multiplying both sides of (12) by 119865(119905 120591)119901(120591) (where119865(119905 120591) is given by (13)) and integrating with respect to 120591 from0 to 119905 and using (5) we get

119905minus120572minus120573minus2120583

Γ (120572)

times int

119905

0

120591120583(119905 minus 120591)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905) 119901 (120591)H (120591 120588) 119889120591

= 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119891 (120588) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

minus 119892 (120588) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) + 119891 (120588) 119892 (120588) 119868

120572120573120578120583

119905119901 (119905)

(14)

Next on multiplying both sides of (14) by 119865(119905 120588)119901(120588) where119865(119905 120588) is given by (13) and integrating with respect to 120588 from0 to 119905 we can write

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 2 (119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905))

(15)

In view of (12) we have

H (120591 120588) = ∬

120588

120591

1198911015840(119910) 1198921015840(119911) 119889119910 119889119911 (16)

Using the followingHolder inequality for the double integral10038161003816100381610038161003816100381610038161003816

∬

120588

120591

119891 (119910) 119892 (119911) 119889119910 119889119911

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

1003816100381610038161003816119891(119910)1003816100381610038161003816

119903119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119904119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(119903minus1+ 119904minus1= 1)

(17)

we obtain

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(18)

4 Abstract and Applied Analysis

Since

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

=1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

=1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

119904minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(19)

then (18) reduces to

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 le

1003816100381610038161003816120591 minus 1205881003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(20)

It follows from (15) that

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591minus120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(21)

Applying againHolderrsquos inequality (17) on the right-hand sideof (21) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120572minus119904120573minus2119904120583

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(22)

In view of the fact that

10038161003816100381610038161003816100381610038161003816

int

120588

120591

1003816100381610038161003816119891(119910)1003816100381610038161003816

119901119889119910

10038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171198911003817100381710038171003817

119901

119901 (23)

we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le [

[

119905minus119903120572minus119903120573minus211990312058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817

119903

119903

Γ119903 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

Abstract and Applied Analysis 5

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119903minus1

times [

[

119905minus119904120572minus119904120573minus211990412058310038171003817100381710038171003817

119892101584010038171003817100381710038171003817

119904

119904

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119904minus1

(24)

From (24) we obtain

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119903minus1

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119904minus1

(25)

Since 119903minus1 + 119904minus1 = 1 the above inequality yields

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(26)

which in view of (15) gives

2100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

(27)

Making use of (26) and (27) the left-hand side of theinequality (11) follows

To prove the right-hand side of the inequality (11) weobserve that 0 le 120591 le 119905 0 le 120588 le 119905 and therefore

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (28)

6 Abstract and Applied Analysis

Evidently from (26) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588) 119889120591 119889120588

=10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(29)

which completes the proof of Theorem 4

The following gives a generalization of Theorem 4

Theorem 5 Let 119901 be a positive function and let 119891 and 119892 betwo synchronous functions on [0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 = 1 then

100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(30)

for all 119905 gt 0 120572 gt max0 minus120573minus120583 120573 lt 1 120583 gt minus1 120573minus1 lt 120578 lt 0120574 gt max0 minus120575 minus ] 120575 lt 1 ] gt minus1 120575 minus 1 lt 120577 lt 0

Proof To prove the above theorem we use inequality (14)Multiplying both sides of (14) by

119905minus120574minus120575minus2]

120588](119905 minus 120588)

120574minus1

Γ (120574)21198651 (120574 + 120575 + ] minus120577 120574 1 minus

120588

119905) 119901 (120588)

(120588 isin (0 119905) 119905 gt 0)

(31)

which remains positive in view of the conditions stated with(30) and then integrating with respect to 120588 from 0 to 119905 we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus 119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

(32)

Now making use of (20) then (32) gives

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

Abstract and Applied Analysis 7

le119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

times

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(33)

Applying Holderrsquos inequality (17) on the right-hand side of(33) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120574minus119904120575minus2119904]

Γ119904 (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + 120583 minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(34)

which on using (23) readily yields the following inequality

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(35)

In view of (32) and (35) and the properties of modulus onecan easily arrive at the left-sided inequality of Theorem 5Moreover we have 0 le 120591 le 119905 0 le 120588 le 119905 hence

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (36)

Therefore from (35) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

8 Abstract and Applied Analysis

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588) 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(37)

which completes the proof of Theorem 5

Remark 6 For 120574 = 120572 120575 = 120573 120577 = 120578 and ] = 120583 Theorem 5immediately reduces to Theorem 4

3 Consequent Results and Special Cases

As implications of our main results we consider someconsequent results of Theorems 4 and 5 by suitably choosingthe function 119901(119905) To this end let us set 119901(119905) = 119905

120582(120582 isin

[0infin) 119905 isin (0infin)) then on using (10) Theorems 4 and 5yield the following results

Corollary 7 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120582 + 120583 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583+120582

120588120583+120582

(119905 minus 120591)120572minus1

(119905 minus 120588)120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 120582 + 1) Γ

2(120582 + 1 minus 120573 + 120578) 119905

1+2120582minus2120573minus2120583

Γ2 (120582 + 1 minus 120573) Γ2 (120583 + 120582 + 1 + 120572 + 120578)

(38)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120582 ge 0min(120582 + 120583 120582 minus 120573 + 120578) gt minus1

Corollary 8 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120574120575120577]119905

119905120582119891 (119905) 119892 (119905)

+Γ (] + 120582 + 1) Γ (120582 + 1 minus 120575 + 120577)

Γ (120582 + 1 minus 120575) Γ (] + 120582 + 1 + 120574 + 120577)119905120582minus120575minus]

times 119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119905120582119891 (119905) 119868

120574120575120577]119905

119905120582119892 (119905)

minus119868120574120575120577]119905

119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574+120575+] minus120577 120574 1minus120588

119905) 120591120583+120582

120588]+120582 1003816100381610038161003816120591minus120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

times (Γ (120583 + 120582 + 1) Γ (] + 120582 + 1)

timesΓ (120582 + 1 minus 120573 + 120578) Γ (120582 + 1 minus 120575 + 120577))

times (Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

timesΓ(120582 + 1 minus 120575)Γ(] + 120582 + 1 + 120574 + 120577))minus1

times 1199051+2120582minus120573minus120575minus2120583minus2]

(39)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0 120582 ge 0min(120582 + 120583 120582 + ] 120582 minus 120573 + 120578 120582 minus 120575 + 120577) gt minus1

Further if we put 120582 = 0 in Corollaries 7 and 8 (or 119901(119905) =1 in Theorems 4 and 5) we obtain the following integralinequalities

Corollary 9 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)119905minus120573minus120583

119868120572120573120578120583

119905119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119891 (119905) 119868

120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

Abstract and Applied Analysis 3

1198921015840isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904

minus1= 1 then (for all 119905 gt 0

120573 lt 1 120583 gt minus1 120572 gt max0 minus120573 minus 120583 120573 minus 1 lt 120578 lt 0)

2100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(11)

Proof Let119891 and 119892 be two synchronous functions then usingDefinition 1 for all 120591 120588 isin (0 119905) 119905 ge 0 we define

H (120591 120588) = (119891 (120591) minus 119891 (120588)) (119892 (120591) minus 119892 (120588)) (12)

Consider

119865 (119905 120591) =119905minus120572minus120573minus2120583

120591120583(119905 minus 120591)

120572minus1

Γ (120572)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

(120591 isin (0 119905) 119905 gt 0)

=120591120583

Γ (120572)

(119905 minus 120591)120572minus1

119905120572+120573+2120583+120591120583(120572 + 120573 + 120583) (minus120578)

Γ (120572 + 1)

(119905 minus 120591)120572

119905120572+120573+2120583+1

+120591120583(120572 + 120573 + 120583) (120572 + 120573 + 120583 + 1) (minus120578) (minus120578 + 1)

2Γ (120572 + 2)

times(119905 minus 120591)

120572+1

119905120572+120573+2120583+2+ sdot sdot sdot

(13)

We observe that each term of the above series is positive inview of the conditions stated withTheorem 4 and hence thefunction 119865(119905 120591) remains positive for all 120591 isin (0 119905) (119905 gt 0)

Multiplying both sides of (12) by 119865(119905 120591)119901(120591) (where119865(119905 120591) is given by (13)) and integrating with respect to 120591 from0 to 119905 and using (5) we get

119905minus120572minus120573minus2120583

Γ (120572)

times int

119905

0

120591120583(119905 minus 120591)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905) 119901 (120591)H (120591 120588) 119889120591

= 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119891 (120588) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

minus 119892 (120588) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) + 119891 (120588) 119892 (120588) 119868

120572120573120578120583

119905119901 (119905)

(14)

Next on multiplying both sides of (14) by 119865(119905 120588)119901(120588) where119865(119905 120588) is given by (13) and integrating with respect to 120588 from0 to 119905 we can write

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 2 (119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905))

(15)

In view of (12) we have

H (120591 120588) = ∬

120588

120591

1198911015840(119910) 1198921015840(119911) 119889119910 119889119911 (16)

Using the followingHolder inequality for the double integral10038161003816100381610038161003816100381610038161003816

∬

120588

120591

119891 (119910) 119892 (119911) 119889119910 119889119911

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

1003816100381610038161003816119891(119910)1003816100381610038161003816

119903119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

1003816100381610038161003816119892 (119911)1003816100381610038161003816

119904119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(119903minus1+ 119904minus1= 1)

(17)

we obtain

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(18)

4 Abstract and Applied Analysis

Since

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

=1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

=1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

119904minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(19)

then (18) reduces to

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 le

1003816100381610038161003816120591 minus 1205881003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(20)

It follows from (15) that

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591minus120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(21)

Applying againHolderrsquos inequality (17) on the right-hand sideof (21) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120572minus119904120573minus2119904120583

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(22)

In view of the fact that

10038161003816100381610038161003816100381610038161003816

int

120588

120591

1003816100381610038161003816119891(119910)1003816100381610038161003816

119901119889119910

10038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171198911003817100381710038171003817

119901

119901 (23)

we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le [

[

119905minus119903120572minus119903120573minus211990312058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817

119903

119903

Γ119903 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

Abstract and Applied Analysis 5

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119903minus1

times [

[

119905minus119904120572minus119904120573minus211990412058310038171003817100381710038171003817

119892101584010038171003817100381710038171003817

119904

119904

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119904minus1

(24)

From (24) we obtain

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119903minus1

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119904minus1

(25)

Since 119903minus1 + 119904minus1 = 1 the above inequality yields

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(26)

which in view of (15) gives

2100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

(27)

Making use of (26) and (27) the left-hand side of theinequality (11) follows

To prove the right-hand side of the inequality (11) weobserve that 0 le 120591 le 119905 0 le 120588 le 119905 and therefore

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (28)

6 Abstract and Applied Analysis

Evidently from (26) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588) 119889120591 119889120588

=10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(29)

which completes the proof of Theorem 4

The following gives a generalization of Theorem 4

Theorem 5 Let 119901 be a positive function and let 119891 and 119892 betwo synchronous functions on [0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 = 1 then

100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(30)

for all 119905 gt 0 120572 gt max0 minus120573minus120583 120573 lt 1 120583 gt minus1 120573minus1 lt 120578 lt 0120574 gt max0 minus120575 minus ] 120575 lt 1 ] gt minus1 120575 minus 1 lt 120577 lt 0

Proof To prove the above theorem we use inequality (14)Multiplying both sides of (14) by

119905minus120574minus120575minus2]

120588](119905 minus 120588)

120574minus1

Γ (120574)21198651 (120574 + 120575 + ] minus120577 120574 1 minus

120588

119905) 119901 (120588)

(120588 isin (0 119905) 119905 gt 0)

(31)

which remains positive in view of the conditions stated with(30) and then integrating with respect to 120588 from 0 to 119905 we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus 119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

(32)

Now making use of (20) then (32) gives

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

Abstract and Applied Analysis 7

le119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

times

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(33)

Applying Holderrsquos inequality (17) on the right-hand side of(33) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120574minus119904120575minus2119904]

Γ119904 (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + 120583 minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(34)

which on using (23) readily yields the following inequality

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(35)

In view of (32) and (35) and the properties of modulus onecan easily arrive at the left-sided inequality of Theorem 5Moreover we have 0 le 120591 le 119905 0 le 120588 le 119905 hence

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (36)

Therefore from (35) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

8 Abstract and Applied Analysis

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588) 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(37)

which completes the proof of Theorem 5

Remark 6 For 120574 = 120572 120575 = 120573 120577 = 120578 and ] = 120583 Theorem 5immediately reduces to Theorem 4

3 Consequent Results and Special Cases

As implications of our main results we consider someconsequent results of Theorems 4 and 5 by suitably choosingthe function 119901(119905) To this end let us set 119901(119905) = 119905

120582(120582 isin

[0infin) 119905 isin (0infin)) then on using (10) Theorems 4 and 5yield the following results

Corollary 7 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120582 + 120583 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583+120582

120588120583+120582

(119905 minus 120591)120572minus1

(119905 minus 120588)120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 120582 + 1) Γ

2(120582 + 1 minus 120573 + 120578) 119905

1+2120582minus2120573minus2120583

Γ2 (120582 + 1 minus 120573) Γ2 (120583 + 120582 + 1 + 120572 + 120578)

(38)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120582 ge 0min(120582 + 120583 120582 minus 120573 + 120578) gt minus1

Corollary 8 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120574120575120577]119905

119905120582119891 (119905) 119892 (119905)

+Γ (] + 120582 + 1) Γ (120582 + 1 minus 120575 + 120577)

Γ (120582 + 1 minus 120575) Γ (] + 120582 + 1 + 120574 + 120577)119905120582minus120575minus]

times 119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119905120582119891 (119905) 119868

120574120575120577]119905

119905120582119892 (119905)

minus119868120574120575120577]119905

119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574+120575+] minus120577 120574 1minus120588

119905) 120591120583+120582

120588]+120582 1003816100381610038161003816120591minus120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

times (Γ (120583 + 120582 + 1) Γ (] + 120582 + 1)

timesΓ (120582 + 1 minus 120573 + 120578) Γ (120582 + 1 minus 120575 + 120577))

times (Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

timesΓ(120582 + 1 minus 120575)Γ(] + 120582 + 1 + 120574 + 120577))minus1

times 1199051+2120582minus120573minus120575minus2120583minus2]

(39)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0 120582 ge 0min(120582 + 120583 120582 + ] 120582 minus 120573 + 120578 120582 minus 120575 + 120577) gt minus1

Further if we put 120582 = 0 in Corollaries 7 and 8 (or 119901(119905) =1 in Theorems 4 and 5) we obtain the following integralinequalities

Corollary 9 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)119905minus120573minus120583

119868120572120573120578120583

119905119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119891 (119905) 119868

120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

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Page 4: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

4 Abstract and Applied Analysis

Since

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119903minus1

=1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

∬

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119910119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

=1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

119904minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(19)

then (18) reduces to

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 le

1003816100381610038161003816120591 minus 1205881003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

(20)

It follows from (15) that

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591minus120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(21)

Applying againHolderrsquos inequality (17) on the right-hand sideof (21) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120572minus119904120573minus2119904120583

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(22)

In view of the fact that

10038161003816100381610038161003816100381610038161003816

int

120588

120591

1003816100381610038161003816119891(119910)1003816100381610038161003816

119901119889119910

10038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171198911003817100381710038171003817

119901

119901 (23)

we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le [

[

119905minus119903120572minus119903120573minus211990312058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817

119903

119903

Γ119903 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

Abstract and Applied Analysis 5

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119903minus1

times [

[

119905minus119904120572minus119904120573minus211990412058310038171003817100381710038171003817

119892101584010038171003817100381710038171003817

119904

119904

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119904minus1

(24)

From (24) we obtain

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119903minus1

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119904minus1

(25)

Since 119903minus1 + 119904minus1 = 1 the above inequality yields

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(26)

which in view of (15) gives

2100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

(27)

Making use of (26) and (27) the left-hand side of theinequality (11) follows

To prove the right-hand side of the inequality (11) weobserve that 0 le 120591 le 119905 0 le 120588 le 119905 and therefore

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (28)

6 Abstract and Applied Analysis

Evidently from (26) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588) 119889120591 119889120588

=10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(29)

which completes the proof of Theorem 4

The following gives a generalization of Theorem 4

Theorem 5 Let 119901 be a positive function and let 119891 and 119892 betwo synchronous functions on [0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 = 1 then

100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(30)

for all 119905 gt 0 120572 gt max0 minus120573minus120583 120573 lt 1 120583 gt minus1 120573minus1 lt 120578 lt 0120574 gt max0 minus120575 minus ] 120575 lt 1 ] gt minus1 120575 minus 1 lt 120577 lt 0

Proof To prove the above theorem we use inequality (14)Multiplying both sides of (14) by

119905minus120574minus120575minus2]

120588](119905 minus 120588)

120574minus1

Γ (120574)21198651 (120574 + 120575 + ] minus120577 120574 1 minus

120588

119905) 119901 (120588)

(120588 isin (0 119905) 119905 gt 0)

(31)

which remains positive in view of the conditions stated with(30) and then integrating with respect to 120588 from 0 to 119905 we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus 119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

(32)

Now making use of (20) then (32) gives

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

Abstract and Applied Analysis 7

le119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

times

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(33)

Applying Holderrsquos inequality (17) on the right-hand side of(33) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120574minus119904120575minus2119904]

Γ119904 (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + 120583 minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(34)

which on using (23) readily yields the following inequality

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(35)

In view of (32) and (35) and the properties of modulus onecan easily arrive at the left-sided inequality of Theorem 5Moreover we have 0 le 120591 le 119905 0 le 120588 le 119905 hence

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (36)

Therefore from (35) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

8 Abstract and Applied Analysis

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588) 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(37)

which completes the proof of Theorem 5

Remark 6 For 120574 = 120572 120575 = 120573 120577 = 120578 and ] = 120583 Theorem 5immediately reduces to Theorem 4

3 Consequent Results and Special Cases

As implications of our main results we consider someconsequent results of Theorems 4 and 5 by suitably choosingthe function 119901(119905) To this end let us set 119901(119905) = 119905

120582(120582 isin

[0infin) 119905 isin (0infin)) then on using (10) Theorems 4 and 5yield the following results

Corollary 7 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120582 + 120583 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583+120582

120588120583+120582

(119905 minus 120591)120572minus1

(119905 minus 120588)120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 120582 + 1) Γ

2(120582 + 1 minus 120573 + 120578) 119905

1+2120582minus2120573minus2120583

Γ2 (120582 + 1 minus 120573) Γ2 (120583 + 120582 + 1 + 120572 + 120578)

(38)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120582 ge 0min(120582 + 120583 120582 minus 120573 + 120578) gt minus1

Corollary 8 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120574120575120577]119905

119905120582119891 (119905) 119892 (119905)

+Γ (] + 120582 + 1) Γ (120582 + 1 minus 120575 + 120577)

Γ (120582 + 1 minus 120575) Γ (] + 120582 + 1 + 120574 + 120577)119905120582minus120575minus]

times 119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119905120582119891 (119905) 119868

120574120575120577]119905

119905120582119892 (119905)

minus119868120574120575120577]119905

119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574+120575+] minus120577 120574 1minus120588

119905) 120591120583+120582

120588]+120582 1003816100381610038161003816120591minus120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

times (Γ (120583 + 120582 + 1) Γ (] + 120582 + 1)

timesΓ (120582 + 1 minus 120573 + 120578) Γ (120582 + 1 minus 120575 + 120577))

times (Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

timesΓ(120582 + 1 minus 120575)Γ(] + 120582 + 1 + 120574 + 120577))minus1

times 1199051+2120582minus120573minus120575minus2120583minus2]

(39)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0 120582 ge 0min(120582 + 120583 120582 + ] 120582 minus 120573 + 120578 120582 minus 120575 + 120577) gt minus1

Further if we put 120582 = 0 in Corollaries 7 and 8 (or 119901(119905) =1 in Theorems 4 and 5) we obtain the following integralinequalities

Corollary 9 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)119905minus120573minus120583

119868120572120573120578120583

119905119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119891 (119905) 119868

120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

Abstract and Applied Analysis 5

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119903minus1

times [

[

119905minus119904120572minus119904120573minus211990412058310038171003817100381710038171003817

119892101584010038171003817100381710038171003817

119904

119904

Γ119904 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

]

119904minus1

(24)

From (24) we obtain

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119903minus1

times [∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588]

119904minus1

(25)

Since 119903minus1 + 119904minus1 = 1 the above inequality yields

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(26)

which in view of (15) gives

2100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

(27)

Making use of (26) and (27) the left-hand side of theinequality (11) follows

To prove the right-hand side of the inequality (11) weobserve that 0 le 120591 le 119905 0 le 120588 le 119905 and therefore

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (28)

6 Abstract and Applied Analysis

Evidently from (26) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588) 119889120591 119889120588

=10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(29)

which completes the proof of Theorem 4

The following gives a generalization of Theorem 4

Theorem 5 Let 119901 be a positive function and let 119891 and 119892 betwo synchronous functions on [0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 = 1 then

100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(30)

for all 119905 gt 0 120572 gt max0 minus120573minus120583 120573 lt 1 120583 gt minus1 120573minus1 lt 120578 lt 0120574 gt max0 minus120575 minus ] 120575 lt 1 ] gt minus1 120575 minus 1 lt 120577 lt 0

Proof To prove the above theorem we use inequality (14)Multiplying both sides of (14) by

119905minus120574minus120575minus2]

120588](119905 minus 120588)

120574minus1

Γ (120574)21198651 (120574 + 120575 + ] minus120577 120574 1 minus

120588

119905) 119901 (120588)

(120588 isin (0 119905) 119905 gt 0)

(31)

which remains positive in view of the conditions stated with(30) and then integrating with respect to 120588 from 0 to 119905 we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus 119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

(32)

Now making use of (20) then (32) gives

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

Abstract and Applied Analysis 7

le119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

times

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(33)

Applying Holderrsquos inequality (17) on the right-hand side of(33) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120574minus119904120575minus2119904]

Γ119904 (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + 120583 minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(34)

which on using (23) readily yields the following inequality

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(35)

In view of (32) and (35) and the properties of modulus onecan easily arrive at the left-sided inequality of Theorem 5Moreover we have 0 le 120591 le 119905 0 le 120588 le 119905 hence

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (36)

Therefore from (35) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

8 Abstract and Applied Analysis

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588) 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(37)

which completes the proof of Theorem 5

Remark 6 For 120574 = 120572 120575 = 120573 120577 = 120578 and ] = 120583 Theorem 5immediately reduces to Theorem 4

3 Consequent Results and Special Cases

As implications of our main results we consider someconsequent results of Theorems 4 and 5 by suitably choosingthe function 119901(119905) To this end let us set 119901(119905) = 119905

120582(120582 isin

[0infin) 119905 isin (0infin)) then on using (10) Theorems 4 and 5yield the following results

Corollary 7 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120582 + 120583 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583+120582

120588120583+120582

(119905 minus 120591)120572minus1

(119905 minus 120588)120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 120582 + 1) Γ

2(120582 + 1 minus 120573 + 120578) 119905

1+2120582minus2120573minus2120583

Γ2 (120582 + 1 minus 120573) Γ2 (120583 + 120582 + 1 + 120572 + 120578)

(38)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120582 ge 0min(120582 + 120583 120582 minus 120573 + 120578) gt minus1

Corollary 8 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120574120575120577]119905

119905120582119891 (119905) 119892 (119905)

+Γ (] + 120582 + 1) Γ (120582 + 1 minus 120575 + 120577)

Γ (120582 + 1 minus 120575) Γ (] + 120582 + 1 + 120574 + 120577)119905120582minus120575minus]

times 119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119905120582119891 (119905) 119868

120574120575120577]119905

119905120582119892 (119905)

minus119868120574120575120577]119905

119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574+120575+] minus120577 120574 1minus120588

119905) 120591120583+120582

120588]+120582 1003816100381610038161003816120591minus120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

times (Γ (120583 + 120582 + 1) Γ (] + 120582 + 1)

timesΓ (120582 + 1 minus 120573 + 120578) Γ (120582 + 1 minus 120575 + 120577))

times (Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

timesΓ(120582 + 1 minus 120575)Γ(] + 120582 + 1 + 120574 + 120577))minus1

times 1199051+2120582minus120573minus120575minus2120583minus2]

(39)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0 120582 ge 0min(120582 + 120583 120582 + ] 120582 minus 120573 + 120578 120582 minus 120575 + 120577) gt minus1

Further if we put 120582 = 0 in Corollaries 7 and 8 (or 119901(119905) =1 in Theorems 4 and 5) we obtain the following integralinequalities

Corollary 9 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)119905minus120573minus120583

119868120572120573120578120583

119905119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119891 (119905) 119868

120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

6 Abstract and Applied Analysis

Evidently from (26) we get

119905minus2120572minus2120573minus4120583

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816H (120591 120588)

1003816100381610038161003816 119889120591 119889120588

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)

times 119901 (120591) 119901 (120588) 119889120591 119889120588

=10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905(119868120572120573120578120583

119905119901(119905))

2

(29)

which completes the proof of Theorem 4

The following gives a generalization of Theorem 4

Theorem 5 Let 119901 be a positive function and let 119891 and 119892 betwo synchronous functions on [0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 = 1 then

100381610038161003816100381610038161003816119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(30)

for all 119905 gt 0 120572 gt max0 minus120573minus120583 120573 lt 1 120583 gt minus1 120573minus1 lt 120578 lt 0120574 gt max0 minus120575 minus ] 120575 lt 1 ] gt minus1 120575 minus 1 lt 120577 lt 0

Proof To prove the above theorem we use inequality (14)Multiplying both sides of (14) by

119905minus120574minus120575minus2]

120588](119905 minus 120588)

120574minus1

Γ (120574)21198651 (120574 + 120575 + ] minus120577 120574 1 minus

120588

119905) 119901 (120588)

(120588 isin (0 119905) 119905 gt 0)

(31)

which remains positive in view of the conditions stated with(30) and then integrating with respect to 120588 from 0 to 119905 we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)H (120591 120588) 119889120591 119889120588

= 119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905) 119891 (119905) 119892 (119905)

+ 119868120574120575120577]119905

119901 (119905) 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119901 (119905) 119891 (119905) 119868

120574120575120577]119905

119901 (119905) 119892 (119905)

minus 119868120574120575120577]119905

119901 (119905) 119891 (119905) 119868120572120573120578120583

119905119901 (119905) 119892 (119905)

(32)

Now making use of (20) then (32) gives

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

Abstract and Applied Analysis 7

le119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

times

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(33)

Applying Holderrsquos inequality (17) on the right-hand side of(33) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120574minus119904120575minus2119904]

Γ119904 (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + 120583 minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(34)

which on using (23) readily yields the following inequality

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(35)

In view of (32) and (35) and the properties of modulus onecan easily arrive at the left-sided inequality of Theorem 5Moreover we have 0 le 120591 le 119905 0 le 120588 le 119905 hence

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (36)

Therefore from (35) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

8 Abstract and Applied Analysis

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588) 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(37)

which completes the proof of Theorem 5

Remark 6 For 120574 = 120572 120575 = 120573 120577 = 120578 and ] = 120583 Theorem 5immediately reduces to Theorem 4

3 Consequent Results and Special Cases

As implications of our main results we consider someconsequent results of Theorems 4 and 5 by suitably choosingthe function 119901(119905) To this end let us set 119901(119905) = 119905

120582(120582 isin

[0infin) 119905 isin (0infin)) then on using (10) Theorems 4 and 5yield the following results

Corollary 7 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120582 + 120583 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583+120582

120588120583+120582

(119905 minus 120591)120572minus1

(119905 minus 120588)120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 120582 + 1) Γ

2(120582 + 1 minus 120573 + 120578) 119905

1+2120582minus2120573minus2120583

Γ2 (120582 + 1 minus 120573) Γ2 (120583 + 120582 + 1 + 120572 + 120578)

(38)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120582 ge 0min(120582 + 120583 120582 minus 120573 + 120578) gt minus1

Corollary 8 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120574120575120577]119905

119905120582119891 (119905) 119892 (119905)

+Γ (] + 120582 + 1) Γ (120582 + 1 minus 120575 + 120577)

Γ (120582 + 1 minus 120575) Γ (] + 120582 + 1 + 120574 + 120577)119905120582minus120575minus]

times 119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119905120582119891 (119905) 119868

120574120575120577]119905

119905120582119892 (119905)

minus119868120574120575120577]119905

119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574+120575+] minus120577 120574 1minus120588

119905) 120591120583+120582

120588]+120582 1003816100381610038161003816120591minus120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

times (Γ (120583 + 120582 + 1) Γ (] + 120582 + 1)

timesΓ (120582 + 1 minus 120573 + 120578) Γ (120582 + 1 minus 120575 + 120577))

times (Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

timesΓ(120582 + 1 minus 120575)Γ(] + 120582 + 1 + 120574 + 120577))minus1

times 1199051+2120582minus120573minus120575minus2120583minus2]

(39)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0 120582 ge 0min(120582 + 120583 120582 + ] 120582 minus 120573 + 120578 120582 minus 120575 + 120577) gt minus1

Further if we put 120582 = 0 in Corollaries 7 and 8 (or 119901(119905) =1 in Theorems 4 and 5) we obtain the following integralinequalities

Corollary 9 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)119905minus120573minus120583

119868120572120573120578120583

119905119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119891 (119905) 119868

120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

Abstract and Applied Analysis 7

le119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119903minus1

times

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119904minus1

119889120591 119889120588

(33)

Applying Holderrsquos inequality (17) on the right-hand side of(33) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le [119905minus119903120572minus119903120573minus2119903120583

Γ119903 (120572)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198911015840(119910)

10038161003816100381610038161003816

119903

119889119910

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119903minus1

times [119905minus119904120574minus119904120575minus2119904]

Γ119904 (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + 120583 minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588)

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

120588

120591

100381610038161003816100381610038161198921015840(119911)

10038161003816100381610038161003816

119904

119889119911

10038161003816100381610038161003816100381610038161003816

119889120591 119889120588]

119904minus1

(34)

which on using (23) readily yields the following inequality

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

times 119901 (120591) 119901 (120588)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

(35)

In view of (32) and (35) and the properties of modulus onecan easily arrive at the left-sided inequality of Theorem 5Moreover we have 0 le 120591 le 119905 0 le 120588 le 119905 hence

0 le1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 le 119905 (36)

Therefore from (35) we get

119905minus120572minus120573minus120574minus120575minus2(120583+])

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905)

8 Abstract and Applied Analysis

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588) 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(37)

which completes the proof of Theorem 5

Remark 6 For 120574 = 120572 120575 = 120573 120577 = 120578 and ] = 120583 Theorem 5immediately reduces to Theorem 4

3 Consequent Results and Special Cases

As implications of our main results we consider someconsequent results of Theorems 4 and 5 by suitably choosingthe function 119901(119905) To this end let us set 119901(119905) = 119905

120582(120582 isin

[0infin) 119905 isin (0infin)) then on using (10) Theorems 4 and 5yield the following results

Corollary 7 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120582 + 120583 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583+120582

120588120583+120582

(119905 minus 120591)120572minus1

(119905 minus 120588)120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 120582 + 1) Γ

2(120582 + 1 minus 120573 + 120578) 119905

1+2120582minus2120573minus2120583

Γ2 (120582 + 1 minus 120573) Γ2 (120583 + 120582 + 1 + 120572 + 120578)

(38)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120582 ge 0min(120582 + 120583 120582 minus 120573 + 120578) gt minus1

Corollary 8 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120574120575120577]119905

119905120582119891 (119905) 119892 (119905)

+Γ (] + 120582 + 1) Γ (120582 + 1 minus 120575 + 120577)

Γ (120582 + 1 minus 120575) Γ (] + 120582 + 1 + 120574 + 120577)119905120582minus120575minus]

times 119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119905120582119891 (119905) 119868

120574120575120577]119905

119905120582119892 (119905)

minus119868120574120575120577]119905

119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574+120575+] minus120577 120574 1minus120588

119905) 120591120583+120582

120588]+120582 1003816100381610038161003816120591minus120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

times (Γ (120583 + 120582 + 1) Γ (] + 120582 + 1)

timesΓ (120582 + 1 minus 120573 + 120578) Γ (120582 + 1 minus 120575 + 120577))

times (Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

timesΓ(120582 + 1 minus 120575)Γ(] + 120582 + 1 + 120574 + 120577))minus1

times 1199051+2120582minus120573minus120575minus2120583minus2]

(39)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0 120582 ge 0min(120582 + 120583 120582 + ] 120582 minus 120573 + 120578 120582 minus 120575 + 120577) gt minus1

Further if we put 120582 = 0 in Corollaries 7 and 8 (or 119901(119905) =1 in Theorems 4 and 5) we obtain the following integralinequalities

Corollary 9 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)119905minus120573minus120583

119868120572120573120578120583

119905119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119891 (119905) 119868

120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

8 Abstract and Applied Analysis

times 120591120583120588]119901 (120591) 119901 (120588)

1003816100381610038161003816H (120591 120588)1003816100381610038161003816 119889120591 119889120588

le

119905minus120572minus120573minus120574minus120575minus2(120583+])10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905

Γ (120572) Γ (120574)

times∬

119905

0

120591120583120588](119905 minus 120591)

120572minus1(119905 minus 120588)

120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 119901 (120591) 119901 (120588) 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904119905119868120572120573120578120583

119905119901 (119905) 119868

120574120575120577]119905

119901 (119905)

(37)

which completes the proof of Theorem 5

Remark 6 For 120574 = 120572 120575 = 120573 120577 = 120578 and ] = 120583 Theorem 5immediately reduces to Theorem 4

3 Consequent Results and Special Cases

As implications of our main results we consider someconsequent results of Theorems 4 and 5 by suitably choosingthe function 119901(119905) To this end let us set 119901(119905) = 119905

120582(120582 isin

[0infin) 119905 isin (0infin)) then on using (10) Theorems 4 and 5yield the following results

Corollary 7 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120582 + 120583 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583+120582

120588120583+120582

(119905 minus 120591)120572minus1

(119905 minus 120588)120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 120582 + 1) Γ

2(120582 + 1 minus 120573 + 120578) 119905

1+2120582minus2120573minus2120583

Γ2 (120582 + 1 minus 120573) Γ2 (120583 + 120582 + 1 + 120572 + 120578)

(38)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120582 ge 0min(120582 + 120583 120582 minus 120573 + 120578) gt minus1

Corollary 8 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 120582 + 1) Γ (120582 + 1 minus 120573 + 120578)

Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

times 119905120582minus120573minus120583

119868120574120575120577]119905

119905120582119891 (119905) 119892 (119905)

+Γ (] + 120582 + 1) Γ (120582 + 1 minus 120575 + 120577)

Γ (120582 + 1 minus 120575) Γ (] + 120582 + 1 + 120574 + 120577)119905120582minus120575minus]

times 119868120572120573120578120583

119905119905120582119891 (119905) 119892 (119905)

minus 119868120572120573120578120583

119905119905120582119891 (119905) 119868

120574120575120577]119905

119905120582119892 (119905)

minus119868120574120575120577]119905

119905120582119891 (119905) 119868

120572120573120578120583

119905119905120582119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574+120575+] minus120577 120574 1minus120588

119905) 120591120583+120582

120588]+120582 1003816100381610038161003816120591minus120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

times (Γ (120583 + 120582 + 1) Γ (] + 120582 + 1)

timesΓ (120582 + 1 minus 120573 + 120578) Γ (120582 + 1 minus 120575 + 120577))

times (Γ (120582 + 1 minus 120573) Γ (120583 + 120582 + 1 + 120572 + 120578)

timesΓ(120582 + 1 minus 120575)Γ(] + 120582 + 1 + 120574 + 120577))minus1

times 1199051+2120582minus120573minus120575minus2120583minus2]

(39)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0 120582 ge 0min(120582 + 120583 120582 + ] 120582 minus 120573 + 120578 120582 minus 120575 + 120577) gt minus1

Further if we put 120582 = 0 in Corollaries 7 and 8 (or 119901(119905) =1 in Theorems 4 and 5) we obtain the following integralinequalities

Corollary 9 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then

2

100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)119905minus120573minus120583

119868120572120573120578120583

119905119891 (119905) 119892 (119905)

minus119868120572120573120578120583

119905119891 (119905) 119868

120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

Abstract and Applied Analysis 9

le

119905minus2120572minus2120573minus412058310038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2 (120572)

times∬

119905

0

120591120583120588120583(119905 minus 120591)

120572minus1(119905 minus 120588)

120572minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120588

119905)1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ2(120583 + 1) Γ

2(1 minus 120573 + 120578) 119905

1minus2120573minus2120583

Γ2 (1 minus 120573) Γ2 (120583 + 1 + 120572 + 120578)

(40)

where 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0

Corollary 10 Let 119891 and 119892 be two synchronous functions on[0infin) If 1198911015840 isin 119871119903([0infin)) 1198921015840 isin 119871 119904([0infin)) 119903 gt 1 119903minus1 + 119904minus1 =1 then100381610038161003816100381610038161003816100381610038161003816

Γ (120583 + 1) Γ (1 minus 120573 + 120578)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578)

times 119905minus120573minus120583

119868120574120575120577]119905

119891 (119905) 119892 (119905)

+Γ (] + 1) Γ (1 minus 120575 + 120577)

Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)119905minus120575minus]

times 119868120572120573120578120583

119905119891 (119905) 119892 (119905) minus 119868

120572120573120578120583

119905119891 (119905) 119868

120574120575120577]119905

119892 (119905)

minus119868120574120575120577]119905

119891 (119905) 119868120572120573120578120583

119905119892 (119905)

100381610038161003816100381610038161003816100381610038161003816

le

119905minus120572minus120573minus120574minus120575minus2120583minus2]10038171003817100381710038171003817

119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

Γ (120572) Γ (120574)

times∬

119905

0

(119905 minus 120591)120572minus1

(119905 minus 120588)120574minus1

times 21198651 (120572 + 120573 + 120583 minus120578 120572 1 minus120591

119905)

times 21198651 (120574 + 120575 + ] minus120577 120574 1 minus120588

119905) 120591120583120588]

times1003816100381610038161003816120591 minus 120588

1003816100381610038161003816 119889120591 119889120588

le10038171003817100381710038171003817119891101584010038171003817100381710038171003817119903

10038171003817100381710038171003817119892101584010038171003817100381710038171003817119904

timesΓ (120583 + 1) Γ (] + 1) Γ (1 minus 120573 + 120578) Γ (1 minus 120575 + 120577)

Γ (1 minus 120573) Γ (120583 + 1 + 120572 + 120578) Γ (1 minus 120575) Γ (] + 1 + 120574 + 120577)

times 1199051minus120573minus120575minus2120583minus2]

(41)

for all 119905 gt 0 120573 lt 1 120583 gt minus1 120572 gt max0 minus120573minus120583 120573minus1 lt 120578 lt 0120575 lt 1 ] gt minus1 120574 gt max0 minus120575 minus ] 120575 minus 1 lt 120577 lt 0

We now briefly consider some consequences of theresults derived in the previous section Following Curiel and

Galue [31] the operator (5) would reduce immediately to theextensively investigated Saigo Erdelyi-Kober and Riemann-Liouville type fractional integral operators respectively givenby the following relationships (see also [30 32])

119868120572120573120578

0119905119891 (119905) = 119868

1205721205731205780

119905119891 (119905)

=119905minus120572minus120573

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

times 21198651 (120572 + 120573 minus120578 120572 1 minus120591

119905)119891 (120591) 119889120591

(120572 gt 0 120573 120578 isin R)

(42)

119868120572120578

119891 (119905) = 11986812057201205780

119905119891 (119905)

=119905minus120572minus120578

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

120591120578119891 (120591) 119889120591

(120572 gt 0 120578 isin R)

(43)

119877120572119891 (119905) = 119868

120572minus1205721205780

119905119891 (119905)

=1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

119891 (120591) 119889120591 (120572 gt 0)

(44)

Now if we consider 120583 = 0 (and ] = 0 additionallyfor Theorem 5) and make use of the relation (42) Theorems4 and 5 provide respectively the known fractional integralinequalities due to Purohit and Raina [29] Again if we set120583 = 0 and replace 120573 by minus120572 (and ] = 0 and replace 120575 by minus120574additionally forTheorem 5) andmake use of the relation (44)then Theorems 4 and 5 correspond to the known integralinequalities due toDahmani et al [28 pages 39ndash42Theorems31 to 32] involving the Riemann-Liouville type fractionalintegral operator

Lastly we conclude this paper by remarking that wehave introduced new general extensions of Chebyshev typeinequalities involving fractional integral hypergeometricoperators By suitably specializing the arbitrary function119901(119905)one can further easily obtain additional integral inequalitiesinvolving the Riemann-Liouville Erdelyi-Kober and Saigotype fractional integral operators from our main results inTheorems 4 and 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to express their thanks to the refereesof the paper for the various suggestions given for the improve-ment of the paper

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

10 Abstract and Applied Analysis

References

[1] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[2] J D Ramırez and A S Vatsala ldquoMonotone iterative techniquefor fractional differential equations with periodic boundaryconditionsrdquo Opuscula Mathematica vol 29 no 3 pp 289ndash3042009

[3] Z Denton and A S Vatsala ldquoMonotone iterative techniquefor finite systems of nonlinear Riemann-Liouville fractionaldifferential equationsrdquoOpusculaMathematica vol 31 no 3 pp327ndash339 2011

[4] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 10 pages 2012

[5] H-R Sun Y-N Li J J Nieto and Q Tang ldquoExistenceof solutions for Sturm-Liouville boundary value problem ofimpulsive differential equationsrdquoAbstract and Applied Analysisvol 2012 Article ID 707163 19 pages 2012

[6] Z-H Zhao Y-K Chang and J J Nieto ldquoAsymptotic behaviorof solutions to abstract stochastic fractional partial integrod-ifferential equationsrdquo Abstract and Applied Analysis vol 2013Article ID 138068 8 pages 2013

[7] Y Liu J J Nieto and Otero-Zarraquinos ldquoExistence results fora coupled system of nonlinear singular fractional differentialequations with impulse effectsrdquo Mathematical Problems inEngineering Article ID 498781 21 pages 2013

[8] P L Chebyshev ldquoSur les expressions approximatives des inte-grales definies par les autres prises entre les memes limitesrdquoProceedings of the Mathematical Society of Kharkov vol 2 pp93ndash98 1882

[9] P Cerone and S S Dragomir ldquoNew upper and lower boundsfor the Cebysev functionalrdquo Journal of Inequalities in Pure andApplied Mathematics vol 3 no 5 article 77 2002

[10] G Gruss ldquoUber das Maximum des absoluten Betrages von

(1(119887minus119886))

119887

int

119886

119891(119909)119892(119909)119889119909minus(1(119887 minus 119886)2)

119887

int

119886

119891(119909)119889119909

119887

int

119886

119892(119909)119889119909rdquoMath-

ematische Zeitschrift vol 39 no 1 pp 215ndash226 1935[11] D SMitrinovic J E Pecaric and AM FinkClassical and New

Inequalities in Analysis vol 61 Kluwer Academic PublishersGroup Dordrecht The Netherlands 1993

[12] G A AnastassiouAdvances on Fractional Inequalities SpringerBriefs in Mathematics Springer New York NY USA 2011

[13] J Pecaric and I Peric ldquoIdentities for the Chebyshev functionalinvolving derivatives of arbitrary order and applicationsrdquo Jour-nal ofMathematical Analysis andApplications vol 313 no 2 pp475ndash483 2006

[14] S Belarbi and Z Dahmani ldquoOn some new fractional integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 10 no 3 article 86 2009

[15] Z Dahmani and L Tabharit ldquoCertain inequalities involvingfractional integralsrdquo Journal of Advanced Research in ScientificComputing vol 2 no 1 pp 55ndash60 2010

[16] S L Kalla and A Rao ldquoOn Gruss type inequality for ahypergeometric fractional integralrdquoLeMatematiche vol 66 no1 pp 57ndash64 2011

[17] H Ogunmez and U M Ozkan ldquoFractional quantum integralinequalitiesrdquo Journal of Inequalities and Applications vol 2011Article ID 787939 7 pages 2011

[18] W T Sulaiman ldquoSome new fractional integral inequalitiesrdquoJournal of Mathematical Analysis vol 2 no 2 pp 23ndash28 2011

[19] M Z Sarikaya and H Ogunmez ldquoOn new inequalities viaRiemann-Liouville fractional integrationrdquoAbstract and AppliedAnalysis vol 2012 Article ID 428983 10 pages 2012

[20] M Z Sarikaya E Set H Yaldiz and N Basak ldquoHermite-Hadamardrsquos inequalities for fractional integrals and relatedfractional inequalitiesrdquoMathematical and Computer Modellingvol 57 no 9-10 pp 2403ndash2407 2013

[21] Z Dahmani and A Benzidane ldquoOn a class of fractional q-integral inequalitiesrdquo Malaya Journal of Matematik vol 3 no1 pp 1ndash6 2013

[22] S D Purohit and R K Raina ldquoChebyshev type inequalities forthe Saigo fractional integrals and their 119902-analoguesrdquo Journal ofMathematical Inequalities vol 7 no 2 pp 239ndash249 2013

[23] S D Purohit F Ucar and R K Yadav ldquoOn fractional integralinequalitiesand their 119902-analoguesrdquoRevista TecnocientificaURUIn press

[24] D Baleanu S D Purohit and P Agarwal ldquoOn fractional inte-gral inequalities involving hypergeometric operatorsrdquo ChineseJournal of Mathematics vol 2014 Article ID 609476 5 pages2014

[25] C Cesarano ldquoIdentities and generating functions onChebyshevpolynomialsrdquo Georgian Mathematical Journal vol 19 no 3 pp427ndash440 2012

[26] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Eliis Horwood Series Mathematical and Its Applica-tions Ellis Horwood Ltd Chichester UK 1984

[27] S S Dragomir ldquoSome integral inequalities of Gruss typerdquoIndian Journal of Pure and Applied Mathematics vol 31 no 4pp 397ndash415 2000

[28] Z Dahmani O Mechouar and S Brahami ldquoCertain inequali-ties related to the Chebyshevrsquos functional involving a Riemann-Liouville operatorrdquoBulletin ofMathematical Analysis and Appli-cations vol 3 no 4 pp 38ndash44 2011

[29] S D Purohit and R K Raina ldquoCertain fractional integralinequalities involving the Gauss hypergeometric functionrdquo

[30] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash143197778

[31] L Curiel and L Galue ldquoA generalization of the integral oper-ators involving the Gaussrsquo hypergeometric functionrdquo RevistaTecnica de la Facultad de Ingenierıa Universidad del Zulia vol19 no 1 pp 17ndash22 1996

[32] V Kiryakova Generalized Fractional Calculus and Applicationsvol 301 of Pitman Research Notes in Mathematics Series Long-man Scientific amp Technical Harlow UK 1994

[33] G E Andrews R Askey and R Roy Special Functions vol 71of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Chebyshev Type Integral Inequalities ...downloads.hindawi.com/journals/aaa/2014/609160.pdfResearch Article Chebyshev Type Integral Inequalities Involving the Fractional

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Research Article New Integral Inequalities with …downloads.hindawi.com/journals/jam/2014/252946.pdfResearch Article New Integral Inequalities with Weakly Singular Kernel for Discontinuous

Research Article New Integral Inequalities with …downloads.hindawi.com/journals/jam/2014/252946.pdfResearch Article New Integral Inequalities with Weakly Singular Kernel for Discontinuous

Integral inequalities Constantin P. Niculescu

Integral inequalities Constantin P. Niculescu

Integral inequalities for algebraic polynomialsgvm/radovi/Integral inequalities for algebraic... · Integral inequalities for algebraic polynomials Gradimir V. Milovanovic Abstract

Integral inequalities for algebraic polynomialsgvm/radovi/Integral inequalities for algebraic... · Integral inequalities for algebraic polynomials Gradimir V. Milovanovic Abstract

Chebyshev Estimator

Chebyshev Estimator

Chebyshev polynomials for solving two dimensional linear ... · In this work, we will apply the Chebyshev polynomials for solving two dimen-sional integral equations of second kind.

Chebyshev polynomials for solving two dimensional linear ... · In this work, we will apply the Chebyshev polynomials for solving two dimen-sional integral equations of second kind.

Chebyshev filter Design

Chebyshev filter Design

Chebyshev Approximations for the Exponential Integral Ei(#)...301 algorithm for rational Chebyshev approximation [11], [12] in 25S arithmetic on a

Chebyshev Approximations for the Exponential Integral Ei(#)...301 algorithm for rational Chebyshev approximation [11], [12] in 25S arithmetic on a

Limit Cycles Investigation for a Class of Nonlinear Systems via Differential and Integral Inequalities

Limit Cycles Investigation for a Class of Nonlinear Systems via Differential and Integral Inequalities

Chebyshev Iteration Methods for Integral Equations … OF COMPUTATION, VOLUME 24, NUMBER 110, APRIL. 1970 Chebyshev Iteration Methods for Integral Equations of the Second Kind By T.

Chebyshev Iteration Methods for Integral Equations … OF COMPUTATION, VOLUME 24, NUMBER 110, APRIL. 1970 Chebyshev Iteration Methods for Integral Equations of the Second Kind By T.

Integral representations and new generating functions of Chebyshev polynomials · 2015-07-30 · Integral representations and new generating functions of Chebyshev polynomials Clemente

Integral representations and new generating functions of Chebyshev polynomials · 2015-07-30 · Integral representations and new generating functions of Chebyshev polynomials Clemente

Chebyshev II

Chebyshev II

Generalization of Integral Inequalities for Functions ... · General Mathematics Vol. 14, No. 4 (2006), 71–96 Generalization of Integral Inequalities for Functions whose Modulus

Generalization of Integral Inequalities for Functions ... · General Mathematics Vol. 14, No. 4 (2006), 71–96 Generalization of Integral Inequalities for Functions whose Modulus

Some modifications in conformable fractional integral inequalities... · 2020. 7. 22. · Baleanuetal.AdvancesinDifferenceEquations20202020:374 Page2of25 betweenconvexityandintegralinequality.Forthisreason,manyknownintegralinequal

Some modifications in conformable fractional integral inequalities... · 2020. 7. 22. · Baleanuetal.AdvancesinDifferenceEquations20202020:374 Page2of25 betweenconvexityandintegralinequality.Forthisreason,manyknownintegralinequal

Some generalization of integral inequalities for twice di ... · Abstract. In this paper, a general integral identity involving Riemann-Liouville fractional integrals is derived.

Some generalization of integral inequalities for twice di ... · Abstract. In this paper, a general integral identity involving Riemann-Liouville fractional integrals is derived.

Chebyshev Coord Transformation Dcaling

Chebyshev Coord Transformation Dcaling

INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS · 2005. 10. 14. · INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS 13 = T(g*+'f-*-'+ι)(Tf)*+q~ι. Raise (11) to the power a, and we have

INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS · 2005. 10. 14. · INEQUALITIES FOR POSITIVE INTEGRAL OPERATORS 13 = T(g*+'f-*-'+ι)(Tf)*+q~ι. Raise (11) to the power a, and we have

Integral Means Inequalities, Convolution, and Univalent ...downloads.hindawi.com/journals/jfs/2019/7817353.pdf · JournalofFunctionSpaces Wereferto[] forthetheoryoftheconvolutionofanalytic

Integral Means Inequalities, Convolution, and Univalent ...downloads.hindawi.com/journals/jfs/2019/7817353.pdf · JournalofFunctionSpaces Wereferto[] forthetheoryoftheconvolutionofanalytic

Numerical Solutions for Weakly Singular Volterra Integral ...ttang/Papers/LiTangXu15.pdf · Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and

Numerical Solutions for Weakly Singular Volterra Integral ...ttang/Papers/LiTangXu15.pdf · Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and

Integral Inequalities of Gronwall-Bellman TypeIntegral Inequalities, One Independent Variable, Partial Differential Equations, Nondecreasing, Nonincreasing 1. Introduction The Gronwall

Integral Inequalities of Gronwall-Bellman TypeIntegral Inequalities, One Independent Variable, Partial Differential Equations, Nondecreasing, Nonincreasing 1. Introduction The Gronwall

chebyshev set

chebyshev set