Research Article Jordan Type Inequalities for...
5
Research Article Jordan Type Inequalities for Hyperbolic Functions and Their Applications Zhen-Hang Yang and Yu-Ming Chu School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China Correspondence should be addressed to Yu-Ming Chu; [email protected] Received 3 August 2014; Accepted 2 September 2014 Academic Editor: Kehe Zhu Copyright © 2015 Z.-H. Yang and Y.-M. Chu. 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. We present the best possible parameters p, ∈ (0, ∞) such that the double inequality (1/3 2 )cosh() + 1 − 1/3 2 < sinh()/ < (1/3 2 )cosh() + 1 − 1/3 2 holds for all ∈ (0,∞). As applications, some new inequalities for certain special function and bivariate means are found. 1. Introduction e well known Jordan inequality [1] is given by 2 < sin () < , ∈ (0, 2 ). (1) During the past few years, the improvements, refine- ments, and generalizations for inequality (1) have attracted the attention of many researchers [2–13]. Recently, the hyperbolic counterpart and its generalizations have been the subject of intensive research. Zhu [14] proved that the inequality ( sinh() ) > + (1 − ) cosh () (2) holds for all > 0 if and only if ≥ 3(1 − ) if ∈ (−∞, 8/15] ∪ (1, ∞). In [3, 15], Neuman and S´ andor proved that cosh 4/3 ( 2 )< sinh () < cosh 3 () (3) for all >0. Kl´ en et al. [5] proved that the double inequality cosh 1/4 () < sinh () < cosh 1/2 () (4) holds for all ∈ (0, 1). In [4], the authors proved that the double inequality cosh () < sinh () < cosh () (5) holds for all ∈ (0, 1) if and only if ≤ 1/3 and ≥ [log(sinh(1))]/[log(cosh(1))] = 0.3721 ⋅ ⋅ ⋅ . Zhu [16, 17] proved that the inequalities (1 − ) + cosh () < ( sinh() ) < (1 − ) + cosh () , ( sinh () ) < (1 − ) + cosh () , + (1 − ) [ coth()−1] <( sinh() ) < + (1 − ) [ coth()−1] (6) hold for all >0 if and only if ≤0, ≥ 1/3, ≤ 1/3, ≥ 1, and ≤ 1/2 if ≥ 4/5, <0, and ≥ 286/693. Very recently, Yang [18] proved that the double inequality [cosh ()] 1/3 2 < sinh () <[cosh ()] 1/3 2 (7) holds for all >0 if and only if ≥ √ 5/5 and ≤ 1/3. Hindawi Publishing Corporation Journal of Function Spaces Volume 2015, Article ID 370979, 4 pages http://dx.doi.org/10.1155/2015/370979
Transcript of Research Article Jordan Type Inequalities for...
Research ArticleJordan Type Inequalities for HyperbolicFunctions and Their Applications
Zhen-Hang Yang and Yu-Ming Chu
School of Mathematics and Computation Sciences Hunan City University Yiyang 413000 China
Correspondence should be addressed to Yu-Ming Chu chuyuming2005126com
Received 3 August 2014 Accepted 2 September 2014
Academic Editor Kehe Zhu
Copyright copy 2015 Z-H Yang and Y-M Chu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We present the best possible parameters p 119902 isin (0infin) such that the double inequality (131199012)cosh(119901119909) + 1 minus 131199012 lt sinh(119909)119909 lt(13119902
2)cosh(119902119909) + 1 minus 13119902
2 holds for all 119909 isin (0infin) As applications some new inequalities for certain special function andbivariate means are found
1 Introduction
The well known Jordan inequality [1] is given by
2
120587119909 lt sin (119909) lt 119909 119909 isin (0
120587
2) (1)
During the past few years the improvements refine-ments and generalizations for inequality (1) have attractedthe attention of many researchers [2ndash13] Recently thehyperbolic counterpart and its generalizations have been thesubject of intensive research
Zhu [14] proved that the inequality
(sinh(119909)119909
)
119902
gt 119901 + (1 minus 119901) cosh (119909) (2)
holds for all 119909 gt 0 if and only if 119902 ge 3(1 minus 119901) if 119901 isin
(minusinfin 815] cup (1infin)In [3 15] Neuman and Sandor proved that
cosh43 (1199092) lt
sinh (119909)119909
lt cosh3 (119909) (3)
for all 119909 gt 0Klen et al [5] proved that the double inequality
cosh14 (119909) lt sinh (119909)119909
lt cosh12 (119909) (4)
holds for all 119909 isin (0 1)
In [4] the authors proved that the double inequality
cosh119901 (119909) lt sinh (119909)119909
lt cosh119902 (119909) (5)
holds for all 119909 isin (0 1) if and only if 119901 le 13 and 119902 ge
[log(sinh(1))][log(cosh(1))] = 03721 sdot sdot sdot Zhu [16 17] proved that the inequalities
(1 minus 120582) + 120582cosh119901 (119909) lt ( sinh(119909)119909
)
119901
lt (1 minus 120583) + 120583cosh119901 (119909)
(sinh (119909)119909
)
119902
lt (1 minus 120578) + 120578cosh119902 (119909)
120572 + (1 minus 120572) 119890119903[119905 coth(119905)minus1]
lt (sinh(119909)119909
)
119903
lt 120573 + (1 minus 120573) 119890119903[119905 coth(119905)minus1]
(6)
hold for all 119909 gt 0 if and only if 120582 le 0 120583 ge 13 120578 le 13 120572 ge1 and 120573 le 12 if 119901 ge 45 119902 lt 0 and 119903 ge 286693
Very recently Yang [18] proved that the double inequality
[cosh (119901119909)]131199012
ltsinh (119909)119909
lt [cosh (119902119909)]131199022
(7)
holds for all 119909 gt 0 if and only if 119901 ge radic55 and 119902 le 13
Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 370979 4 pageshttpdxdoiorg1011552015370979
2 Journal of Function Spaces
The main purpose of this paper is to find the bestpossible parameters 119901 119902 isin (0infin) such that the doubleinequality (131199012) cosh(119901119909) + 1 minus 131199012 lt sinh(119909)119909 lt
(131199022) cosh(119902119909) + 1 minus 13119902
2 holds for all 119909 gt 0 andpresent several new inequalities for certain special functionand bivariate means
2 Main Result
Theorem 1 Let 119901 119902 isin (0infin) Then the double inequality
1
31199012cosh (119901119909) + 1 minus 1
31199012ltsinh (119909)119909
lt1
31199022cosh (119902119909) + 1 minus 1
31199022
(8)
holds for all 119909 gt 0 if and only if 119901 le radic155 and 119902 ge 1
Proof Let 120582 gt 0 and let the function 119891120582be defined on (0infin)
by
119891120582(119909) =
sinh (119909)119909
minus (1
31205822cosh (120582119909) + 1 minus 1
31205822) (9)
Then making use of power series expansions and (9) we get
119891120582(119909) =
infin
sum
119899=0
1199092119899
(2119899 + 1)minus (
1
31205822
infin
sum
119899=0
(120582119909)2119899
(2119899)+ 1 minus
1
31205822)
=
infin
sum
119899=2
3 minus (2119899 + 1) 1205822119899minus2
3 (2119899 + 1)1199092119899
(10)
Let
119886119899(120582) = 3 minus (2119899 + 1) 120582
2119899minus2 (11)
Then
119886119899(1) = 3 minus (2119899 + 1) = minus2 (119899 minus 1) lt 0 (12)
for all 119899 ge 2 Consider
1198862(radic15
5) = 0 119886
3(radic15
5) =
12
25gt 0
119886119899+1(radic15
5) minus 119886119899(radic15
5) =
4 times 3119899minus1
5119899(119899 minus 1) gt 0
(13)
for all 119899 ge 2It follows from (13) that
1198862(radic15
5) = 0 119886
119899(radic15
5) gt 0 (14)
for all 119899 ge 3Therefore inequality (8) holds for all 119909 gt 0 with 119901 =
radic155 and 119902 = 1 follows from (9)ndash(12) and (14)Next we prove that 119901 le radic155 and 119902 ge 1 if inequality (8)
holds for all 119909 gt 0
If the first inequality of (8) holds for all 119909 gt 0 then from(9) and (10) we have
lim119909rarr0
+
119891119901(119909)
1199094=3 minus 5119901
2
360ge 0 (15)
and 119901 le radic155If the second inequality of (8) holds for all 119909 gt 0 then it
follows from (9) that
lim119909rarr+infin
119891119902(119909)
119890119902119909=1 minus 119890minus2119909
2119909119890(1minus119902)119909
minus1 + 119890minus2119902119909
61199022
minus (1 minus1
31199022) 119890minus119902119909
le 0
(16)
We clearly see that lim119909rarr+infin
(119891119902(119909)119890119902119909) = +infin if 119902 lt 1
Therefore 119902 ge 1 follows from (16)
Remark 2 It is not difficult to verify that the bound
119892119901(119909) =
1
31199012cosh (119901119909) + 1 minus 1
31199012(17)
given in Theorem 1 is strictly increasing with respect to 119901 on(0infin) for fixed 119909 isin (0infin)
Remark 3 Let 119901 = radic155 gt 34 gt radic22 gt 23 gt radic33 and119902 = 1 lt 2radic33 ThenTheorem 1 and Remark 2 lead to
cosh(radic3119909
3) lt
3
4cosh (2119909
3) +
1
4lt2
3cosh(
radic2119909
2) +
1
3
lt16
27cosh (3119909
4) +
11
27lt5
9cosh(
radic15119909
5)
+4
9ltsinh (119909)119909
lt1
3cosh (119909) + 2
3lt1
2cosh2 (
radic3119909
3) +
1
2
(18)
for all 119909 gt 0
3 Applications
It is well known that
int
infin
0
119909
sinh (119909)=1
21205951015840(1
2) (19)
where 1205951015840 is the trigamma function defined by
1205951015840(119909) = int
infin
0
119905119890minus119909119905
1 minus 119890minus119905119889119905 (20)
Let
Sh (119909) = int119909
0
119905
sinh (119905)119889119905 (21)
Journal of Function Spaces 3
Then Remark 3 leads to
int
119909
0
3
cosh (119905) + 2119889119905 lt Sh (119909) lt int
119909
0
9
5 cosh (radic151199055) + 4119889119905
(22)for all 119909 gt 0
From (19) and (22) we get the following
Remark 4 For all 119909 gt 0 one has
radic3 log(119890119909+ 2 minus radic3
119890119909 + 2 + radic3) + radic3 log (2 + radic3)
lt Sh (119909) lt 2radic15
times [arctan(53119890radic151199095
+4
3) minus arctan (3)]
(23)In particular we have
45620 sdot sdot sdot = 2radic3 log (2 + radic3) lt 1205951015840 (12)
lt 2radic15 [120587 minus 2 arctan (3)] = 49845 sdot sdot sdot (24)
For 119886 119887 gt 0 the Schwab-Borchardt mean SB(119886 119887) [19ndash21]is given by
SB (119886 119887) =radic1198872 minus 1198862
arccos (119886119887)(119886 lt 119887)
SB (119886 119887) = 119886 (119886 = 119887) (25)
SB (119886 119887) =radic1198862 minus 1198872
coshminus1 (119886119887)(119886 gt 119887) (26)
Let 119886 gt 119887 and let 119909 = coshminus1(119886119887) Then cosh(119909) = 119886119887and radic1198862 minus 1198872coshminus1(119886119887) = 119887 sinh(119909)119909 It follows fromRemark 3 and (26) that
[16
27cosh (3119909
4) +
11
27] 119887 lt SB (119886 119887) lt [1
3cosh (119909) + 2
3] 119887
(27)Note that
[1
3cosh (119909) + 2
3] 119887 =
119886 + 2119887
3
[16
27cosh (3119909
4) +
11
27] 119887
=811988714
27(radic2(119886 + 119887)(2119886 minus 119887)
2+ 211988732)
12
+11119887
27
(28)From (27) and (28) we get the following
Remark 5 Let 119886 gt 119887 gt 0 then the Schwab-Borchardt meanSB(119886 119887) satisfies the double inequality
811988714
27(radic2(119886 + 119887) (2119886 minus 119887)
2+ 211988732)
12
+11119887
27lt SB (119886 119887) lt 119886 + 2119887
3
(29)
Let 119901 isin R and let 119886 119887 gt 0 with 119886 = 119887 Then the arith-metic mean 119860(119886 119887) logarithmic mean 119871(119886 119887) geometricmean 119866(119886 119887) and 119901th power mean119872
119901(119886 119887) are defined by
119860 (119886 119887) =119886 + 119887
2 119871 (119886 119887) =
119887 minus 119886
log 119887 minus log 119886
119866 (119886 119887) = radic119886119887
119872119901(119886 119887) = (
119886119901+ 119887119901
2)
1119901
(119901 = 0)
1198720(119886 119887) = radic119886119887 = 119866 (119886 119887)
(30)
It is well known that119872119901(119886 119887) is continuous and strictly
increasing with respect to 119901 isin R for fixed 119886 119887 gt 0 with119886 = 119887 the main properties for the power mean are givenin [22] Recently the arithmetic logarithmic geometric andpower means have been the subject of intensive research Inparticular many remarkable inequalities can be found in theliterature [23ndash35]
Let 119909 = (12) log(119886119887) then (30) leads to
sinh (119909)119909
=119871 (119886 119887)
119866 (119886 119887) cosh (119901119909) = (
119872119901(119886 119887)
119866(119886 119887))
119901
(31)
FromTheorem 1 and (31) we get the following
Remark 6 Let 119901 119902 isin (0infin) then the double inequality
1
31199012119872119901
119901(119886 119887) 119866
1minus119901(119886 119887) + (1 minus
1
31199012)119866 (119886 119887)
lt 119871 (119886 119887) lt1
31199022119872119902
119902(119886 119887) 119866
1minus119902(119886 119887)
+ (1 minus1
31199022)119866 (119886 119887)
(32)
holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 119901 le radic155 and119902 ge 1 In particular the double inequality
5
9119872radic155
radic155(119886 119887) 119866
1minusradic155(119886 119887) +
4
9119866 (119886 119887)
lt 119871 (119886 119887) lt1
3119860 (119886 119887) +
2
3119866 (119886 119887)
(33)
holds for all 119886 119887 gt 0 with 119886 = 119887
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 61374086 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrant LY13A010004
4 Journal of Function Spaces
References
[1] D S Mitrinovic Analytic Inequalities Springer New York NYUSA 1970
[2] F Qi L-H Cui and S-L Xu ldquoSome inequalities constructedby Tchebysheffs integral inequalityrdquo Mathematical Inequalitiesand Applications vol 2 no 4 pp 517ndash528 1999
[3] E Neuman and J Sandor ldquoOn some inequalities involvingtrigonometric and hyperbolic functions with emphasis on theCusa-Huygens Wilker and Huygens inequalitiesrdquo Mathemati-cal Inequalities amp Applications vol 13 no 4 pp 715ndash723 2010
[4] Y Lv G Wang and Y Chu ldquoA note on Jordan type inequalitiesfor hyperbolic functionsrdquo Applied Mathematics Letters vol 25no 3 pp 505ndash508 2012
[5] R Klen M Visuri and M Vuorinen ldquoOn Jordan typeinequalities for hyperbolic functionsrdquo Journal of Inequalities andApplications vol 2010 Article ID 362548 14 pages 2010
[6] Z-H Yang ldquoNew sharp Jordan type inequalities and theirapplicationsrdquo Gulf Journal of Mathematics vol 2 no 1 pp 1ndash10 2014
[7] L Zhu ldquoA source of inequalities for circular functionsrdquoComput-ers amp Mathematics with Applications vol 58 no 10 pp 1998ndash2004 2009
[8] Z-H Yang ldquoSharp bounds for seiffert mean in terms ofweighted power means of arithmetic mean and geometricmeanrdquo Mathematical Inequalities and Applications vol 17 no2 pp 499ndash511 2014
[9] Z-H Yang Y-M Chu Y-Q Song and Y-M Li ldquoA sharp dou-ble inequality for trigonometric functions and its applicationsrdquoAbstract and Applied Analysis vol 2014 Article ID 592085 9pages 2014
[10] S-HWu and L Debnath ldquoA new generalized and sharp versionof Jordanrsquos inequality and its applications to the improvement ofthe Yang Le inequalityrdquo Applied Mathematics Letters vol 19 no12 pp 1378ndash1384 2006
[11] S-H Wu ldquoSharpness and generalization of Jordanrsquos inequalityand its applicationrdquo Taiwanese Journal of Mathematics vol 12no 2 pp 325ndash336 2008
[12] Z Yang ldquoRefinements of a two-sided inequality for trigonomet-ric functionsrdquo Journal of Mathematical Inequalities vol 7 no 4pp 601ndash615 2013
[13] Z Yang and Y Chu ldquoA note on Jordan Adamovic-Mitrinovicand Cusa Inequalitiesrdquo Abstract and Applied Analysis vol 2014Article ID 364076 12 pages 2014
[14] L Zhu ldquoGeneralized Lazarevicrsquos inequality and its applicationsIIrdquo Journal of Inequalities and Applications vol 2009 Article ID379142 4 pages 2009
[15] E Neuman and J Sandor ldquoInequalities for hyperbolic func-tionsrdquo Applied Mathematics and Computation vol 218 no 18pp 9291ndash9295 2012
[16] L Zhu ldquoInequalities for hyperbolic functions and their applica-tionsrdquo Journal of Inequalities and Applications vol 2010 ArticleID 130821 10 pages 2010
[17] L Zhu ldquoNew inequalities for hyperbolic functions and theirapplicationsrdquo Journal of Inequalities and Applications vol 2012article 303 9 pages 2012
[18] Z-H Yang ldquoNew sharp bounds for logarithmic mean andidentric meanrdquo Journal of Inequalities and Applications vol2013 article 116 2013
[19] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[20] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[21] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[22] P S Bullen D S Mitrinovic and P M Vasic Means andTheir Inequalities vol 31 D Reidel Publishing Co DordrechtNetherlands 1988
[23] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 pp 615ndash618 1972
[24] T P Lin ldquoThe power mean and the logarithmic meanrdquo TheAmerican Mathematical Monthly vol 81 pp 879ndash883 1974
[25] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 2-3 pp 261ndash270 1990
[26] H Alzer and S Qiu ldquoInequalities for means in two variablesrdquoArchiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[27] Y-M Chu and W-F Xia ldquoTwo optimal double inequalitiesbetween power mean and logarithmic meanrdquo Computers ampMathematics with Applications vol 60 no 1 pp 83ndash89 2010
[28] Y Chu Y Qiu and M Wang ldquoSharp power mean bounds forthe combination of Seiffert and geometric meansrdquo Abstract andApplied Analysis vol 2010 Article ID 108920 12 pages 2010
[29] H Liu and X-J Meng ldquoThe optimal convex combinationbounds for Seiffertrsquos meanrdquo Journal of Inequalities and Applica-tions vol 2011 Article ID 686834 9 pages 2011
[30] H Gao J Guo and W Yu ldquoSharp bounds for power meanin terms of generalized Heronian meanrdquo Abstract and AppliedAnalysis vol 2011 Article ID 679201 9 pages 2011
[31] M-K Wang Y-M Chu Y-F Qiu and S-L Qiu ldquoAn optimalpower mean inequality for the complete elliptic integralsrdquoApplied Mathematics Letters vol 24 no 6 pp 887ndash890 2011
[32] W-D Jiang and F Qi ldquoSome sharp inequalities involving Seif-fert and other means and their concise proofsrdquo MathematicalInequalities amp Applications vol 15 no 4 Article ID 15-86 pp1007ndash1017 2012
[33] W-D Jiang ldquoSome sharp inequalities involving reciprocalsof the Seiffert and other meansrdquo Journal of MathematicalInequalities vol 6 no 4 pp 593ndash599 2012
[34] Y M Chu M Y Shi and Y P Jiang ldquoOptimal inequalitiesfor the power harmonic and logarithmic meansrdquo IranianMathematical Society Bulletin vol 38 no 3 pp 597ndash606 2012
[35] Y M Chu S L Qiu and M K Wang ldquoSharp inequalitiesinvolving the power mean and complete elliptic integral of thefirst kindrdquoThe Rocky Mountain Journal of Mathematics vol 43no 5 pp 1489ndash1496 2013
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
The main purpose of this paper is to find the bestpossible parameters 119901 119902 isin (0infin) such that the doubleinequality (131199012) cosh(119901119909) + 1 minus 131199012 lt sinh(119909)119909 lt
(131199022) cosh(119902119909) + 1 minus 13119902
2 holds for all 119909 gt 0 andpresent several new inequalities for certain special functionand bivariate means
2 Main Result
Theorem 1 Let 119901 119902 isin (0infin) Then the double inequality
1
31199012cosh (119901119909) + 1 minus 1
31199012ltsinh (119909)119909
lt1
31199022cosh (119902119909) + 1 minus 1
31199022
(8)
holds for all 119909 gt 0 if and only if 119901 le radic155 and 119902 ge 1
Proof Let 120582 gt 0 and let the function 119891120582be defined on (0infin)
by
119891120582(119909) =
sinh (119909)119909
minus (1
31205822cosh (120582119909) + 1 minus 1
31205822) (9)
Then making use of power series expansions and (9) we get
119891120582(119909) =
infin
sum
119899=0
1199092119899
(2119899 + 1)minus (
1
31205822
infin
sum
119899=0
(120582119909)2119899
(2119899)+ 1 minus
1
31205822)
=
infin
sum
119899=2
3 minus (2119899 + 1) 1205822119899minus2
3 (2119899 + 1)1199092119899
(10)
Let
119886119899(120582) = 3 minus (2119899 + 1) 120582
2119899minus2 (11)
Then
119886119899(1) = 3 minus (2119899 + 1) = minus2 (119899 minus 1) lt 0 (12)
for all 119899 ge 2 Consider
1198862(radic15
5) = 0 119886
3(radic15
5) =
12
25gt 0
119886119899+1(radic15
5) minus 119886119899(radic15
5) =
4 times 3119899minus1
5119899(119899 minus 1) gt 0
(13)
for all 119899 ge 2It follows from (13) that
1198862(radic15
5) = 0 119886
119899(radic15
5) gt 0 (14)
for all 119899 ge 3Therefore inequality (8) holds for all 119909 gt 0 with 119901 =
radic155 and 119902 = 1 follows from (9)ndash(12) and (14)Next we prove that 119901 le radic155 and 119902 ge 1 if inequality (8)
holds for all 119909 gt 0
If the first inequality of (8) holds for all 119909 gt 0 then from(9) and (10) we have
lim119909rarr0
+
119891119901(119909)
1199094=3 minus 5119901
2
360ge 0 (15)
and 119901 le radic155If the second inequality of (8) holds for all 119909 gt 0 then it
follows from (9) that
lim119909rarr+infin
119891119902(119909)
119890119902119909=1 minus 119890minus2119909
2119909119890(1minus119902)119909
minus1 + 119890minus2119902119909
61199022
minus (1 minus1
31199022) 119890minus119902119909
le 0
(16)
We clearly see that lim119909rarr+infin
(119891119902(119909)119890119902119909) = +infin if 119902 lt 1
Therefore 119902 ge 1 follows from (16)
Remark 2 It is not difficult to verify that the bound
119892119901(119909) =
1
31199012cosh (119901119909) + 1 minus 1
31199012(17)
given in Theorem 1 is strictly increasing with respect to 119901 on(0infin) for fixed 119909 isin (0infin)
Remark 3 Let 119901 = radic155 gt 34 gt radic22 gt 23 gt radic33 and119902 = 1 lt 2radic33 ThenTheorem 1 and Remark 2 lead to
cosh(radic3119909
3) lt
3
4cosh (2119909
3) +
1
4lt2
3cosh(
radic2119909
2) +
1
3
lt16
27cosh (3119909
4) +
11
27lt5
9cosh(
radic15119909
5)
+4
9ltsinh (119909)119909
lt1
3cosh (119909) + 2
3lt1
2cosh2 (
radic3119909
3) +
1
2
(18)
for all 119909 gt 0
3 Applications
It is well known that
int
infin
0
119909
sinh (119909)=1
21205951015840(1
2) (19)
where 1205951015840 is the trigamma function defined by
1205951015840(119909) = int
infin
0
119905119890minus119909119905
1 minus 119890minus119905119889119905 (20)
Let
Sh (119909) = int119909
0
119905
sinh (119905)119889119905 (21)
Journal of Function Spaces 3
Then Remark 3 leads to
int
119909
0
3
cosh (119905) + 2119889119905 lt Sh (119909) lt int
119909
0
9
5 cosh (radic151199055) + 4119889119905
(22)for all 119909 gt 0
From (19) and (22) we get the following
Remark 4 For all 119909 gt 0 one has
radic3 log(119890119909+ 2 minus radic3
119890119909 + 2 + radic3) + radic3 log (2 + radic3)
lt Sh (119909) lt 2radic15
times [arctan(53119890radic151199095
+4
3) minus arctan (3)]
(23)In particular we have
45620 sdot sdot sdot = 2radic3 log (2 + radic3) lt 1205951015840 (12)
lt 2radic15 [120587 minus 2 arctan (3)] = 49845 sdot sdot sdot (24)
For 119886 119887 gt 0 the Schwab-Borchardt mean SB(119886 119887) [19ndash21]is given by
SB (119886 119887) =radic1198872 minus 1198862
arccos (119886119887)(119886 lt 119887)
SB (119886 119887) = 119886 (119886 = 119887) (25)
SB (119886 119887) =radic1198862 minus 1198872
coshminus1 (119886119887)(119886 gt 119887) (26)
Let 119886 gt 119887 and let 119909 = coshminus1(119886119887) Then cosh(119909) = 119886119887and radic1198862 minus 1198872coshminus1(119886119887) = 119887 sinh(119909)119909 It follows fromRemark 3 and (26) that
[16
27cosh (3119909
4) +
11
27] 119887 lt SB (119886 119887) lt [1
3cosh (119909) + 2
3] 119887
(27)Note that
[1
3cosh (119909) + 2
3] 119887 =
119886 + 2119887
3
[16
27cosh (3119909
4) +
11
27] 119887
=811988714
27(radic2(119886 + 119887)(2119886 minus 119887)
2+ 211988732)
12
+11119887
27
(28)From (27) and (28) we get the following
Remark 5 Let 119886 gt 119887 gt 0 then the Schwab-Borchardt meanSB(119886 119887) satisfies the double inequality
811988714
27(radic2(119886 + 119887) (2119886 minus 119887)
2+ 211988732)
12
+11119887
27lt SB (119886 119887) lt 119886 + 2119887
3
(29)
Let 119901 isin R and let 119886 119887 gt 0 with 119886 = 119887 Then the arith-metic mean 119860(119886 119887) logarithmic mean 119871(119886 119887) geometricmean 119866(119886 119887) and 119901th power mean119872
119901(119886 119887) are defined by
119860 (119886 119887) =119886 + 119887
2 119871 (119886 119887) =
119887 minus 119886
log 119887 minus log 119886
119866 (119886 119887) = radic119886119887
119872119901(119886 119887) = (
119886119901+ 119887119901
2)
1119901
(119901 = 0)
1198720(119886 119887) = radic119886119887 = 119866 (119886 119887)
(30)
It is well known that119872119901(119886 119887) is continuous and strictly
increasing with respect to 119901 isin R for fixed 119886 119887 gt 0 with119886 = 119887 the main properties for the power mean are givenin [22] Recently the arithmetic logarithmic geometric andpower means have been the subject of intensive research Inparticular many remarkable inequalities can be found in theliterature [23ndash35]
Let 119909 = (12) log(119886119887) then (30) leads to
sinh (119909)119909
=119871 (119886 119887)
119866 (119886 119887) cosh (119901119909) = (
119872119901(119886 119887)
119866(119886 119887))
119901
(31)
FromTheorem 1 and (31) we get the following
Remark 6 Let 119901 119902 isin (0infin) then the double inequality
1
31199012119872119901
119901(119886 119887) 119866
1minus119901(119886 119887) + (1 minus
1
31199012)119866 (119886 119887)
lt 119871 (119886 119887) lt1
31199022119872119902
119902(119886 119887) 119866
1minus119902(119886 119887)
+ (1 minus1
31199022)119866 (119886 119887)
(32)
holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 119901 le radic155 and119902 ge 1 In particular the double inequality
5
9119872radic155
radic155(119886 119887) 119866
1minusradic155(119886 119887) +
4
9119866 (119886 119887)
lt 119871 (119886 119887) lt1
3119860 (119886 119887) +
2
3119866 (119886 119887)
(33)
holds for all 119886 119887 gt 0 with 119886 = 119887
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 61374086 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrant LY13A010004
4 Journal of Function Spaces
References
[1] D S Mitrinovic Analytic Inequalities Springer New York NYUSA 1970
[2] F Qi L-H Cui and S-L Xu ldquoSome inequalities constructedby Tchebysheffs integral inequalityrdquo Mathematical Inequalitiesand Applications vol 2 no 4 pp 517ndash528 1999
[3] E Neuman and J Sandor ldquoOn some inequalities involvingtrigonometric and hyperbolic functions with emphasis on theCusa-Huygens Wilker and Huygens inequalitiesrdquo Mathemati-cal Inequalities amp Applications vol 13 no 4 pp 715ndash723 2010
[4] Y Lv G Wang and Y Chu ldquoA note on Jordan type inequalitiesfor hyperbolic functionsrdquo Applied Mathematics Letters vol 25no 3 pp 505ndash508 2012
[5] R Klen M Visuri and M Vuorinen ldquoOn Jordan typeinequalities for hyperbolic functionsrdquo Journal of Inequalities andApplications vol 2010 Article ID 362548 14 pages 2010
[6] Z-H Yang ldquoNew sharp Jordan type inequalities and theirapplicationsrdquo Gulf Journal of Mathematics vol 2 no 1 pp 1ndash10 2014
[7] L Zhu ldquoA source of inequalities for circular functionsrdquoComput-ers amp Mathematics with Applications vol 58 no 10 pp 1998ndash2004 2009
[8] Z-H Yang ldquoSharp bounds for seiffert mean in terms ofweighted power means of arithmetic mean and geometricmeanrdquo Mathematical Inequalities and Applications vol 17 no2 pp 499ndash511 2014
[9] Z-H Yang Y-M Chu Y-Q Song and Y-M Li ldquoA sharp dou-ble inequality for trigonometric functions and its applicationsrdquoAbstract and Applied Analysis vol 2014 Article ID 592085 9pages 2014
[10] S-HWu and L Debnath ldquoA new generalized and sharp versionof Jordanrsquos inequality and its applications to the improvement ofthe Yang Le inequalityrdquo Applied Mathematics Letters vol 19 no12 pp 1378ndash1384 2006
[11] S-H Wu ldquoSharpness and generalization of Jordanrsquos inequalityand its applicationrdquo Taiwanese Journal of Mathematics vol 12no 2 pp 325ndash336 2008
[12] Z Yang ldquoRefinements of a two-sided inequality for trigonomet-ric functionsrdquo Journal of Mathematical Inequalities vol 7 no 4pp 601ndash615 2013
[13] Z Yang and Y Chu ldquoA note on Jordan Adamovic-Mitrinovicand Cusa Inequalitiesrdquo Abstract and Applied Analysis vol 2014Article ID 364076 12 pages 2014
[14] L Zhu ldquoGeneralized Lazarevicrsquos inequality and its applicationsIIrdquo Journal of Inequalities and Applications vol 2009 Article ID379142 4 pages 2009
[15] E Neuman and J Sandor ldquoInequalities for hyperbolic func-tionsrdquo Applied Mathematics and Computation vol 218 no 18pp 9291ndash9295 2012
[16] L Zhu ldquoInequalities for hyperbolic functions and their applica-tionsrdquo Journal of Inequalities and Applications vol 2010 ArticleID 130821 10 pages 2010
[17] L Zhu ldquoNew inequalities for hyperbolic functions and theirapplicationsrdquo Journal of Inequalities and Applications vol 2012article 303 9 pages 2012
[18] Z-H Yang ldquoNew sharp bounds for logarithmic mean andidentric meanrdquo Journal of Inequalities and Applications vol2013 article 116 2013
[19] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[20] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[21] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[22] P S Bullen D S Mitrinovic and P M Vasic Means andTheir Inequalities vol 31 D Reidel Publishing Co DordrechtNetherlands 1988
[23] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 pp 615ndash618 1972
[24] T P Lin ldquoThe power mean and the logarithmic meanrdquo TheAmerican Mathematical Monthly vol 81 pp 879ndash883 1974
[25] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 2-3 pp 261ndash270 1990
[26] H Alzer and S Qiu ldquoInequalities for means in two variablesrdquoArchiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[27] Y-M Chu and W-F Xia ldquoTwo optimal double inequalitiesbetween power mean and logarithmic meanrdquo Computers ampMathematics with Applications vol 60 no 1 pp 83ndash89 2010
[28] Y Chu Y Qiu and M Wang ldquoSharp power mean bounds forthe combination of Seiffert and geometric meansrdquo Abstract andApplied Analysis vol 2010 Article ID 108920 12 pages 2010
[29] H Liu and X-J Meng ldquoThe optimal convex combinationbounds for Seiffertrsquos meanrdquo Journal of Inequalities and Applica-tions vol 2011 Article ID 686834 9 pages 2011
[30] H Gao J Guo and W Yu ldquoSharp bounds for power meanin terms of generalized Heronian meanrdquo Abstract and AppliedAnalysis vol 2011 Article ID 679201 9 pages 2011
[31] M-K Wang Y-M Chu Y-F Qiu and S-L Qiu ldquoAn optimalpower mean inequality for the complete elliptic integralsrdquoApplied Mathematics Letters vol 24 no 6 pp 887ndash890 2011
[32] W-D Jiang and F Qi ldquoSome sharp inequalities involving Seif-fert and other means and their concise proofsrdquo MathematicalInequalities amp Applications vol 15 no 4 Article ID 15-86 pp1007ndash1017 2012
[33] W-D Jiang ldquoSome sharp inequalities involving reciprocalsof the Seiffert and other meansrdquo Journal of MathematicalInequalities vol 6 no 4 pp 593ndash599 2012
[34] Y M Chu M Y Shi and Y P Jiang ldquoOptimal inequalitiesfor the power harmonic and logarithmic meansrdquo IranianMathematical Society Bulletin vol 38 no 3 pp 597ndash606 2012
[35] Y M Chu S L Qiu and M K Wang ldquoSharp inequalitiesinvolving the power mean and complete elliptic integral of thefirst kindrdquoThe Rocky Mountain Journal of Mathematics vol 43no 5 pp 1489ndash1496 2013
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
Journal of Function Spaces 3
Then Remark 3 leads to
int
119909
0
3
cosh (119905) + 2119889119905 lt Sh (119909) lt int
119909
0
9
5 cosh (radic151199055) + 4119889119905
(22)for all 119909 gt 0
From (19) and (22) we get the following
Remark 4 For all 119909 gt 0 one has
radic3 log(119890119909+ 2 minus radic3
119890119909 + 2 + radic3) + radic3 log (2 + radic3)
lt Sh (119909) lt 2radic15
times [arctan(53119890radic151199095
+4
3) minus arctan (3)]
(23)In particular we have
45620 sdot sdot sdot = 2radic3 log (2 + radic3) lt 1205951015840 (12)
lt 2radic15 [120587 minus 2 arctan (3)] = 49845 sdot sdot sdot (24)
For 119886 119887 gt 0 the Schwab-Borchardt mean SB(119886 119887) [19ndash21]is given by
SB (119886 119887) =radic1198872 minus 1198862
arccos (119886119887)(119886 lt 119887)
SB (119886 119887) = 119886 (119886 = 119887) (25)
SB (119886 119887) =radic1198862 minus 1198872
coshminus1 (119886119887)(119886 gt 119887) (26)
Let 119886 gt 119887 and let 119909 = coshminus1(119886119887) Then cosh(119909) = 119886119887and radic1198862 minus 1198872coshminus1(119886119887) = 119887 sinh(119909)119909 It follows fromRemark 3 and (26) that
[16
27cosh (3119909
4) +
11
27] 119887 lt SB (119886 119887) lt [1
3cosh (119909) + 2
3] 119887
(27)Note that
[1
3cosh (119909) + 2
3] 119887 =
119886 + 2119887
3
[16
27cosh (3119909
4) +
11
27] 119887
=811988714
27(radic2(119886 + 119887)(2119886 minus 119887)
2+ 211988732)
12
+11119887
27
(28)From (27) and (28) we get the following
Remark 5 Let 119886 gt 119887 gt 0 then the Schwab-Borchardt meanSB(119886 119887) satisfies the double inequality
811988714
27(radic2(119886 + 119887) (2119886 minus 119887)
2+ 211988732)
12
+11119887
27lt SB (119886 119887) lt 119886 + 2119887
3
(29)
Let 119901 isin R and let 119886 119887 gt 0 with 119886 = 119887 Then the arith-metic mean 119860(119886 119887) logarithmic mean 119871(119886 119887) geometricmean 119866(119886 119887) and 119901th power mean119872
119901(119886 119887) are defined by
119860 (119886 119887) =119886 + 119887
2 119871 (119886 119887) =
119887 minus 119886
log 119887 minus log 119886
119866 (119886 119887) = radic119886119887
119872119901(119886 119887) = (
119886119901+ 119887119901
2)
1119901
(119901 = 0)
1198720(119886 119887) = radic119886119887 = 119866 (119886 119887)
(30)
It is well known that119872119901(119886 119887) is continuous and strictly
increasing with respect to 119901 isin R for fixed 119886 119887 gt 0 with119886 = 119887 the main properties for the power mean are givenin [22] Recently the arithmetic logarithmic geometric andpower means have been the subject of intensive research Inparticular many remarkable inequalities can be found in theliterature [23ndash35]
Let 119909 = (12) log(119886119887) then (30) leads to
sinh (119909)119909
=119871 (119886 119887)
119866 (119886 119887) cosh (119901119909) = (
119872119901(119886 119887)
119866(119886 119887))
119901
(31)
FromTheorem 1 and (31) we get the following
Remark 6 Let 119901 119902 isin (0infin) then the double inequality
1
31199012119872119901
119901(119886 119887) 119866
1minus119901(119886 119887) + (1 minus
1
31199012)119866 (119886 119887)
lt 119871 (119886 119887) lt1
31199022119872119902
119902(119886 119887) 119866
1minus119902(119886 119887)
+ (1 minus1
31199022)119866 (119886 119887)
(32)
holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 119901 le radic155 and119902 ge 1 In particular the double inequality
5
9119872radic155
radic155(119886 119887) 119866
1minusradic155(119886 119887) +
4
9119866 (119886 119887)
lt 119871 (119886 119887) lt1
3119860 (119886 119887) +
2
3119866 (119886 119887)
(33)
holds for all 119886 119887 gt 0 with 119886 = 119887
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 61374086 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrant LY13A010004
4 Journal of Function Spaces
References
[1] D S Mitrinovic Analytic Inequalities Springer New York NYUSA 1970
[2] F Qi L-H Cui and S-L Xu ldquoSome inequalities constructedby Tchebysheffs integral inequalityrdquo Mathematical Inequalitiesand Applications vol 2 no 4 pp 517ndash528 1999
[3] E Neuman and J Sandor ldquoOn some inequalities involvingtrigonometric and hyperbolic functions with emphasis on theCusa-Huygens Wilker and Huygens inequalitiesrdquo Mathemati-cal Inequalities amp Applications vol 13 no 4 pp 715ndash723 2010
[4] Y Lv G Wang and Y Chu ldquoA note on Jordan type inequalitiesfor hyperbolic functionsrdquo Applied Mathematics Letters vol 25no 3 pp 505ndash508 2012
[5] R Klen M Visuri and M Vuorinen ldquoOn Jordan typeinequalities for hyperbolic functionsrdquo Journal of Inequalities andApplications vol 2010 Article ID 362548 14 pages 2010
[6] Z-H Yang ldquoNew sharp Jordan type inequalities and theirapplicationsrdquo Gulf Journal of Mathematics vol 2 no 1 pp 1ndash10 2014
[7] L Zhu ldquoA source of inequalities for circular functionsrdquoComput-ers amp Mathematics with Applications vol 58 no 10 pp 1998ndash2004 2009
[8] Z-H Yang ldquoSharp bounds for seiffert mean in terms ofweighted power means of arithmetic mean and geometricmeanrdquo Mathematical Inequalities and Applications vol 17 no2 pp 499ndash511 2014
[9] Z-H Yang Y-M Chu Y-Q Song and Y-M Li ldquoA sharp dou-ble inequality for trigonometric functions and its applicationsrdquoAbstract and Applied Analysis vol 2014 Article ID 592085 9pages 2014
[10] S-HWu and L Debnath ldquoA new generalized and sharp versionof Jordanrsquos inequality and its applications to the improvement ofthe Yang Le inequalityrdquo Applied Mathematics Letters vol 19 no12 pp 1378ndash1384 2006
[11] S-H Wu ldquoSharpness and generalization of Jordanrsquos inequalityand its applicationrdquo Taiwanese Journal of Mathematics vol 12no 2 pp 325ndash336 2008
[12] Z Yang ldquoRefinements of a two-sided inequality for trigonomet-ric functionsrdquo Journal of Mathematical Inequalities vol 7 no 4pp 601ndash615 2013
[13] Z Yang and Y Chu ldquoA note on Jordan Adamovic-Mitrinovicand Cusa Inequalitiesrdquo Abstract and Applied Analysis vol 2014Article ID 364076 12 pages 2014
[14] L Zhu ldquoGeneralized Lazarevicrsquos inequality and its applicationsIIrdquo Journal of Inequalities and Applications vol 2009 Article ID379142 4 pages 2009
[15] E Neuman and J Sandor ldquoInequalities for hyperbolic func-tionsrdquo Applied Mathematics and Computation vol 218 no 18pp 9291ndash9295 2012
[16] L Zhu ldquoInequalities for hyperbolic functions and their applica-tionsrdquo Journal of Inequalities and Applications vol 2010 ArticleID 130821 10 pages 2010
[17] L Zhu ldquoNew inequalities for hyperbolic functions and theirapplicationsrdquo Journal of Inequalities and Applications vol 2012article 303 9 pages 2012
[18] Z-H Yang ldquoNew sharp bounds for logarithmic mean andidentric meanrdquo Journal of Inequalities and Applications vol2013 article 116 2013
[19] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[20] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[21] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[22] P S Bullen D S Mitrinovic and P M Vasic Means andTheir Inequalities vol 31 D Reidel Publishing Co DordrechtNetherlands 1988
[23] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 pp 615ndash618 1972
[24] T P Lin ldquoThe power mean and the logarithmic meanrdquo TheAmerican Mathematical Monthly vol 81 pp 879ndash883 1974
[25] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 2-3 pp 261ndash270 1990
[26] H Alzer and S Qiu ldquoInequalities for means in two variablesrdquoArchiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[27] Y-M Chu and W-F Xia ldquoTwo optimal double inequalitiesbetween power mean and logarithmic meanrdquo Computers ampMathematics with Applications vol 60 no 1 pp 83ndash89 2010
[28] Y Chu Y Qiu and M Wang ldquoSharp power mean bounds forthe combination of Seiffert and geometric meansrdquo Abstract andApplied Analysis vol 2010 Article ID 108920 12 pages 2010
[29] H Liu and X-J Meng ldquoThe optimal convex combinationbounds for Seiffertrsquos meanrdquo Journal of Inequalities and Applica-tions vol 2011 Article ID 686834 9 pages 2011
[30] H Gao J Guo and W Yu ldquoSharp bounds for power meanin terms of generalized Heronian meanrdquo Abstract and AppliedAnalysis vol 2011 Article ID 679201 9 pages 2011
[31] M-K Wang Y-M Chu Y-F Qiu and S-L Qiu ldquoAn optimalpower mean inequality for the complete elliptic integralsrdquoApplied Mathematics Letters vol 24 no 6 pp 887ndash890 2011
[32] W-D Jiang and F Qi ldquoSome sharp inequalities involving Seif-fert and other means and their concise proofsrdquo MathematicalInequalities amp Applications vol 15 no 4 Article ID 15-86 pp1007ndash1017 2012
[33] W-D Jiang ldquoSome sharp inequalities involving reciprocalsof the Seiffert and other meansrdquo Journal of MathematicalInequalities vol 6 no 4 pp 593ndash599 2012
[34] Y M Chu M Y Shi and Y P Jiang ldquoOptimal inequalitiesfor the power harmonic and logarithmic meansrdquo IranianMathematical Society Bulletin vol 38 no 3 pp 597ndash606 2012
[35] Y M Chu S L Qiu and M K Wang ldquoSharp inequalitiesinvolving the power mean and complete elliptic integral of thefirst kindrdquoThe Rocky Mountain Journal of Mathematics vol 43no 5 pp 1489ndash1496 2013
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
4 Journal of Function Spaces
References
[1] D S Mitrinovic Analytic Inequalities Springer New York NYUSA 1970
[2] F Qi L-H Cui and S-L Xu ldquoSome inequalities constructedby Tchebysheffs integral inequalityrdquo Mathematical Inequalitiesand Applications vol 2 no 4 pp 517ndash528 1999
[3] E Neuman and J Sandor ldquoOn some inequalities involvingtrigonometric and hyperbolic functions with emphasis on theCusa-Huygens Wilker and Huygens inequalitiesrdquo Mathemati-cal Inequalities amp Applications vol 13 no 4 pp 715ndash723 2010
[4] Y Lv G Wang and Y Chu ldquoA note on Jordan type inequalitiesfor hyperbolic functionsrdquo Applied Mathematics Letters vol 25no 3 pp 505ndash508 2012
[5] R Klen M Visuri and M Vuorinen ldquoOn Jordan typeinequalities for hyperbolic functionsrdquo Journal of Inequalities andApplications vol 2010 Article ID 362548 14 pages 2010
[6] Z-H Yang ldquoNew sharp Jordan type inequalities and theirapplicationsrdquo Gulf Journal of Mathematics vol 2 no 1 pp 1ndash10 2014
[7] L Zhu ldquoA source of inequalities for circular functionsrdquoComput-ers amp Mathematics with Applications vol 58 no 10 pp 1998ndash2004 2009
[8] Z-H Yang ldquoSharp bounds for seiffert mean in terms ofweighted power means of arithmetic mean and geometricmeanrdquo Mathematical Inequalities and Applications vol 17 no2 pp 499ndash511 2014
[9] Z-H Yang Y-M Chu Y-Q Song and Y-M Li ldquoA sharp dou-ble inequality for trigonometric functions and its applicationsrdquoAbstract and Applied Analysis vol 2014 Article ID 592085 9pages 2014
[10] S-HWu and L Debnath ldquoA new generalized and sharp versionof Jordanrsquos inequality and its applications to the improvement ofthe Yang Le inequalityrdquo Applied Mathematics Letters vol 19 no12 pp 1378ndash1384 2006
[11] S-H Wu ldquoSharpness and generalization of Jordanrsquos inequalityand its applicationrdquo Taiwanese Journal of Mathematics vol 12no 2 pp 325ndash336 2008
[12] Z Yang ldquoRefinements of a two-sided inequality for trigonomet-ric functionsrdquo Journal of Mathematical Inequalities vol 7 no 4pp 601ndash615 2013
[13] Z Yang and Y Chu ldquoA note on Jordan Adamovic-Mitrinovicand Cusa Inequalitiesrdquo Abstract and Applied Analysis vol 2014Article ID 364076 12 pages 2014
[14] L Zhu ldquoGeneralized Lazarevicrsquos inequality and its applicationsIIrdquo Journal of Inequalities and Applications vol 2009 Article ID379142 4 pages 2009
[15] E Neuman and J Sandor ldquoInequalities for hyperbolic func-tionsrdquo Applied Mathematics and Computation vol 218 no 18pp 9291ndash9295 2012
[16] L Zhu ldquoInequalities for hyperbolic functions and their applica-tionsrdquo Journal of Inequalities and Applications vol 2010 ArticleID 130821 10 pages 2010
[17] L Zhu ldquoNew inequalities for hyperbolic functions and theirapplicationsrdquo Journal of Inequalities and Applications vol 2012article 303 9 pages 2012
[18] Z-H Yang ldquoNew sharp bounds for logarithmic mean andidentric meanrdquo Journal of Inequalities and Applications vol2013 article 116 2013
[19] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[20] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[21] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[22] P S Bullen D S Mitrinovic and P M Vasic Means andTheir Inequalities vol 31 D Reidel Publishing Co DordrechtNetherlands 1988
[23] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 pp 615ndash618 1972
[24] T P Lin ldquoThe power mean and the logarithmic meanrdquo TheAmerican Mathematical Monthly vol 81 pp 879ndash883 1974
[25] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 2-3 pp 261ndash270 1990
[26] H Alzer and S Qiu ldquoInequalities for means in two variablesrdquoArchiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[27] Y-M Chu and W-F Xia ldquoTwo optimal double inequalitiesbetween power mean and logarithmic meanrdquo Computers ampMathematics with Applications vol 60 no 1 pp 83ndash89 2010
[28] Y Chu Y Qiu and M Wang ldquoSharp power mean bounds forthe combination of Seiffert and geometric meansrdquo Abstract andApplied Analysis vol 2010 Article ID 108920 12 pages 2010
[29] H Liu and X-J Meng ldquoThe optimal convex combinationbounds for Seiffertrsquos meanrdquo Journal of Inequalities and Applica-tions vol 2011 Article ID 686834 9 pages 2011
[30] H Gao J Guo and W Yu ldquoSharp bounds for power meanin terms of generalized Heronian meanrdquo Abstract and AppliedAnalysis vol 2011 Article ID 679201 9 pages 2011
[31] M-K Wang Y-M Chu Y-F Qiu and S-L Qiu ldquoAn optimalpower mean inequality for the complete elliptic integralsrdquoApplied Mathematics Letters vol 24 no 6 pp 887ndash890 2011
[32] W-D Jiang and F Qi ldquoSome sharp inequalities involving Seif-fert and other means and their concise proofsrdquo MathematicalInequalities amp Applications vol 15 no 4 Article ID 15-86 pp1007ndash1017 2012
[33] W-D Jiang ldquoSome sharp inequalities involving reciprocalsof the Seiffert and other meansrdquo Journal of MathematicalInequalities vol 6 no 4 pp 593ndash599 2012
[34] Y M Chu M Y Shi and Y P Jiang ldquoOptimal inequalitiesfor the power harmonic and logarithmic meansrdquo IranianMathematical Society Bulletin vol 38 no 3 pp 597ndash606 2012
[35] Y M Chu S L Qiu and M K Wang ldquoSharp inequalitiesinvolving the power mean and complete elliptic integral of thefirst kindrdquoThe Rocky Mountain Journal of Mathematics vol 43no 5 pp 1489ndash1496 2013
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
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
