MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World...

8
Research Article On a New Class of -Valent Meromorphic Functions Defined in Conic Domains Mohammed Ali Alamri and Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia Correspondence should be addressed to Maslina Darus; [email protected] Received 26 April 2016; Accepted 27 June 2016 Academic Editor: Jin-Lin Liu Copyright © 2016 M. A. Alamri and M. Darus. 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 define a new class of multivalent meromorphic functions using the generalised hypergeometric function. We derived this class related to conic domain. It is also shown that this new class of functions, under certain conditions, becomes a class of starlike functions. Some results on inclusion and closure properties are also derived. 1. Introduction Let denote the class of functions of the form () = 1 + =1 , N = 1, 2, 3, . . . , (1) which are analytic and -valent in the punctured unit disc centred at origin = { : 0 < || < 1} = \ {0}. Also by () ≺ () we mean () is subordinate to () which implies the existence of an analytic function, called Schwartz function () with |()| < 1, for such that () = (()), where () and () are multivalent meromorphic functions. Note that if is univalent in then the above subordination is equivalent to (0) = (0) and () ⊂ (). e set of points, for 0<<1 and ∈ [0, ∞), Ω , = Ω + (1 − ) , (2) where Ω = { + V :> ( − 1) 2 + V 2 , > 0} , (3) [1] showed that the extremal functions , () for conic regions are convex univalent and given by , () = { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { { 1 + (1 − 2) 1− , = 0, 1+ 2 1− 2 [ 2 (arccos ) arctanh ] , 0 < < 1, 1+ 2 2 (log 1+√ 1−√ ) 2 , = 1, 1+ 2 −1 [ [ [ sin ( 2 () ()/√ 0 1 1− 2 1 − () 2 d) − 1 ] ] ] , > 1, (4) where () is Legendre’s complete elliptic integral of the first kind with () = 1− 2 as its complementary integral, () = ( − √ )/(1 − √ ), ∈ (0, 1), and is chosen in such a way that = cosh( ()/()). Hindawi Publishing Corporation e Scientific World Journal Volume 2016, Article ID 6360250, 7 pages http://dx.doi.org/10.1155/2016/6360250

Transcript of MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World...

Page 1: MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World Journal andcomparingthecoe cientsof 2 gives 2 s 2 ti1+T 1 j= 2 i1+T 1 jsV V (1

Research ArticleOn a New Class of 119901-Valent Meromorphic FunctionsDefined in Conic Domains

Mohammed Ali Alamri and Maslina Darus

School ofMathematical Sciences Faculty of Science and Technology Universiti KebangsaanMalaysia 43600 Bangi SelangorMalaysia

Correspondence should be addressed to Maslina Darus maslinaukmedumy

Received 26 April 2016 Accepted 27 June 2016

Academic Editor Jin-Lin Liu

Copyright copy 2016 M A Alamri and M Darus 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 define a new class of multivalent meromorphic functions using the generalised hypergeometric function We derived this classrelated to conic domain It is also shown that this new class of functions under certain conditions becomes a class of starlikefunctions Some results on inclusion and closure properties are also derived

1 Introduction

Let119872119901 denote the class of functions of the form

119891 (119911) =1

119911119901+

infin

sum

119899=1

119886119899119911119899minus119901 119901 isin N = 1 2 3 (1)

which are analytic and 119901-valent in the punctured unit disccentred at origin 119864lowast = 119911 0 lt |119911| lt 1 = 119864 0Also by 119891(119911) ≺ 119892(119911) we mean 119891(119911) is subordinate to 119892(119911)which implies the existence of an analytic function calledSchwartz function 119908(119911) with |119908(119911)| lt 1 for 119911 isin 119864lowast such that119891(119911) = 119892(119908(119911)) where 119891(119911) and 119892(119911) are multivalent

meromorphic functions Note that if 119892 is univalent in 119864 thenthe above subordination is equivalent to 119891(0) = 119892(0) and119891(119864) sub 119892(119864)

The set of points for 0 lt 120574 lt 1 and 119896 isin [0infin)

Ω119896120574 = 120574Ω119896 + (1 minus 120574) (2)

where

Ω119896 = 119906 + 119894V 119906 gt 119896radic(119906 minus 1)2+ V2 119906 gt 0 (3)

[1] showed that the extremal functions 119902119896120574(119911) for conicregions are convex univalent and given by

119902119896120574 (119911) =

1 + (1 minus 2120574) 119911

1 minus 119911 119896 = 0

1 +2120574

1 minus 1198962[2

120587(arccos 119896) arctanhradic119911] 0 lt 119896 lt 1

1 +2120574

1205872(log 1 +

radic119911

1 minus radic119911)

2

119896 = 1

1 +120574

1198962 minus 1

[[

[

sin( 120587

2119877 (119905)int

119906(119911)radic119905

0

1

radic1 minus 1199092radic1 minus (119905119909)2

d119909) minus 1]]

]

119896 gt 1

(4)

where 119877(119905) is Legendrersquos complete elliptic integral of the firstkind with 1198771015840(119905) = radic1 minus 1199052 as its complementary integral

119906(119911) = (119911 minus radic119905)(1 minus radic119905119911) 119905 isin (0 1) and 119911 isin 119864 is chosenin such a way that 119896 = cosh(1205871198771015840(119905)119877(119905))

Hindawi Publishing Corporatione Scientific World JournalVolume 2016 Article ID 6360250 7 pageshttpdxdoiorg10115520166360250

2 The Scientific World Journal

The generalised hypergeometric function119902119865119904(1205721 120572119902

1205731 120573119904 119911) for complex parameters1205721 120572119902 and1205731 120573119904with 120573119895 = 0 minus1 minus2 minus3 for 119895 = 1 2 3 119904 is defined as

119902119865119904 (1205721 120572119902 1205731 120573119904 119911) =

infin

sum

119899=0

(1205721)119899sdot sdot sdot (120572119902)119899

(1205731)119899sdot sdot sdot (120573119904)119899

119899119911119899

= 1 +

infin

sum

119899=1

(1205721)119899sdot sdot sdot (120572119902)119899

(1205731)119899sdot sdot sdot (120573119904)119899

119911119899

(119899)

(5)

with 119902 le 119904+1 119902 119904 isin N0 = Ncup0 and (120572)119899 is the well-knownPochhammer symbol related to the factorial and the Gammafunction by the relation

(120572)119899 =(120572 + 119899 minus 1)

(120572 minus 1)=Γ (120572 + 119899)

Γ (120572) (6)

Also (5) implies

ℎ119901 (1205721 120572119902 1205731 120573119904 119911)

= 119911minus119901

119902119865119904 (1205721 120572119902 1205731 120573119904 119911) isin 119872119901

(7)

Liu and Srivastava [2] defined a linear operator for func-tions belonging to the class of multivalent meromorphicfunction119867119901(1205721 120572119902 1205731 120573119904) 119872119901 rarr119872119901 as follows

119867119901 (1205721 120572119902 1205731 120573119904)

= ℎ119901 (1205721 120572119902 1205731 120573119904 119911) ⋆ 119891 (119911)

(8)

If we assume for brevity that 119867119901119902119904(1205721) = 119867119901(1205721 120572119902

1205731 120573119904) then the following identity holds for this operator

119911 [119867119901119902119904 (1205721) 119891 (119911)]1015840

= 1205721119867119901119902119904 (1205721 + 1) 119891 (119911)

minus (1205721 + 119901)119867119901119902119904 (1205721) 119891 (119911)

(9)

Shareef [3] defined and studied subclass119872119876119901(119896 120582 1205721) ofmeromorphic function associated with conic domain for 119896 ge0 0 le 120582 lt 1 and 119901 ge 1 as follows

minus1

119901

[

[

119911 (119867119901119902119904 (1205721) 119891 (119911))1015840

+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840]

]

≺ 119902119896120574 (119911)

(10)

We now define a new subclass119872119876119901(119887 119896 120582 1205721) of mero-morphic function associated with conic domain for 119896 ge 00 le 120582 lt 1 119901 ge 1 and 119887 ge 1 as follows

minus1

119901

[

[

1

+1

119887(

119911 (119867119901119902119904 (1205721) 119891 (119911))1015840

+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840)

minus 119887]

]

≺ 119902119896120574 (119911)

(11)

Since 119902119896120574 is a convex and univalent function for ℎ(119911) ≺119902119896120574(119911) it means ℎ(119864lowast) is contained in 119902119896120574(119864

lowast) where

ℎ (119911) = minus1

119901

[

[

1

+1

119887(

119911 (119867119901119902119904 (1205721) 119891 (119911))1015840

+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840)

minus 119887]

]

(12)

In the next two sections for brevity we drop the subscripts ofthe operator119867119901119902119904(1205721)

2 Preliminary Results

Lemma 1 (see [4]) Let ℎ2(119911) be convex in 119864 andR(120582ℎ2(119911) +120583) gt 0 where 120583 isin C 120582 isin C0 and 119911 isin 119864 If ℎ1(119911) is analyticin 119864 with ℎ1(0) = ℎ2(0) then

ℎ1 (119911) +119911ℎ1015840

1(119911)

120582ℎ1 (119911) + 120583≺ ℎ2 (119911)

119894119898119901119897119894119890119904 ℎ1 (119911) ≺ ℎ2 (119911)

(13)

Lemma 2 (see [5]) Let ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 and 119867(119911) =

1 + suminfin

119899=1119889119899119911119899 and ℎ ≺ 119867 If 119867(119911) is univalent and convex in

119864 then10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611988911003816100381610038161003816 (14)

for 119899 ge 1

Lemma 3 (see [6]) If 119902119896120574(119911) = 1 + 1199021119911 + 11990221199112 + sdot sdot sdot then

1199021 =

8 (1 minus 120574) (arccos (119896))2

1205872 (1 minus 1198962) 0 lt 119896 lt 1

8 (1 minus 120574)

1205872 119896 = 1

1205872(1 minus 120574)

4radic119905 (1198962 minus 1) 1198962 (119905) (1 + 119905) 119896 gt 1

(15)

One now states and proves the main results

3 Main Results

In this section we explore some of the geometric propertiesexhibited by the class119872119876119901(119887 119896 120582 1205721)

We begin by discussing an inclusion property for the class119872119876119901(119887 119896 120582 1205721)

Theorem 4 IfR(1205721) gt 119901R(119887119902119896120582(119911) minus 1) then

119872119876119901 (119887 119896 120582 1205721 + 1) sub 119872119876119901 (119887 119896 120582 1205721) (16)

The Scientific World Journal 3

Proof Let 119891 isin 119872119876119901(119887 119896 120582 1205721 + 1) and set

minus1

119901[1

+1

119887(119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840)

minus 119887] = ℎ (119911)

(17)

But differentiating (9) with respect to 119911 we get

[119911119867 (1205721) 119891 (119911)]1015840

= 1205721119867((1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840

119911 [119867 (1205721) 119891 (119911)]10158401015840+ [119867 (1205721) 119891 (119911)]

1015840

= 1205721119867((1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840

1199112(119867 (1205721) 119891 (119911))

10158401015840

= 1205721119911 (119867 (1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901 + 1) 119911 (119867 (1205721) 119891 (119911))1015840

(18)

Putting (18) in (17) we have

1205821199111205721 (119867 (120572 + 1) 119891 (119911))1015840+ [1 minus 120582 (1205721 + 119901 + 1)] 119911 (119867 (1205721) 119891 (119911))

1015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= minus119887119901ℎ (119911)

1205821205721119911 (119867 (1205721 + 1) 119891 (119911))1015840+ (1 minus 120582) 1205721 (119867 (1205721 + 1) 119891 (119911))

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= minus119887119901ℎ (119911) + (1205721 + 119901)

(19)

Taking logarithmic derivative of (19) we have

119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840

+ 119887119901ℎ (119911) + 119887 (119887 minus 1) =minus119887119901119911ℎ

1015840(119911)

minus119887119901ℎ (119911) + 1205721 + 119901

119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840

= minus119887119901[ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + 1205721 + 119901] minus 119887 (119887 minus 1)

minus1

119901[1

+1

119887(119911 (119867 (1205721 + 1) 119891 (119911))

1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721 + 1) 119891 (119911) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840)

minus 119887] = ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + (1205721 + 119901)

(20)

Since 119891 isin 119872119876119902(119887 119896 120582 1205721 + 1) therefore

ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + (1205721 + 119901)≺ 119902119896120574 (119911) (21)

Using Lemma 2 we have

ℎ (119911) ≺ 119902119896120574 (119911) (22)

providedR(minus119887119901119902119896120582(119911)+1205721 +119901) gt 0 or equivalentlyR(1205721) gt119901R(119887119902119896120574(119911) minus 1) Hence 119891 isin 119872119876119901(119887 119896 120582 1205721)

We now show that the class 119872119876119901(119887 119896 120582 1205721) is closedunder a certain integral

Theorem 5 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then the integral

119861120578 (119867 (120572) 119891 (119911)) =120578 minus 119901

119911120578int

119911

0

119905120578minus1(119867 (120572) 119891 (119911)) d119905 (23)

maps 119891(119911) into119872119876119901(119887 119896 120582 1205721)

Proof From (23) we have

119911120578119861120578 (119867 (120572) 119891 (119911)) = (120578 minus 119901)int

119911

0

119905120578minus1(119867 (120572) 119891 (119911)) d119905 (24)

Note that

119861120578 (119867 (120572) 119891 (119911)) = (119867 (120572)) 119861120578119891 (119911) (25)

Differentiating (24) above we get

120578119911120578minus1119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))

1015840

= (120578 minus 119901) 119911120578minus1(119867 (120572) 119891 (119911))

4 The Scientific World Journal

120578119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))1015840

= (120578 minus 119901) (119867 (120572) 119891 (119911))

(26)

Differentiate again

120578 (119861120578 (119867 (120572) 119891 (119911)))1015840

+ 119911 (119861120578 (119867 (120572) 119891 (119911)))10158401015840

+ (119861120578 (119867 (120572) 119891 (119911)))1015840

= (120578 minus 119901) (119867 (120572) 119891 (119911))1015840

119911 (119861120578 (119891 (119911)))10158401015840

= (120578 minus 119901) (119861120578 (119867 (120572) 119891 (119911)))1015840

minus (120578 + 1) (119861120578 (119891 (119911)))1015840

(27)

Now let

minus1

119901

[

[

1

+1

119887(

119911 (119861120578 (119867 (1205721) 119891))1015840

+ 1205821199112(119861120578 (119867 (1205721) 119891))

10158401015840

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840)

minus 119887]

]

= 119892 (119911)

(28)

Using (26) and (27) in (28) we get

119911119861120578 (119867 (1205721) 119891)1015840+ 120582 [119911 (120578 minus 119901) (119867 (1205721) 119891)

1015840minus 119911 (1 + 120578) (119861120578 (119867 (1205721) 119891))

1015840

]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

120582119911 (120578 minus 119901) (119867 (1205721) 119891)1015840+ [1 minus 120582 (1 + 120578)] [(120578 minus 119901) (119867 (1205721) 119891) minus 120578 (119861120578 (119867 (1205721) 119891))]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)] minus 120582120578119911 (119861120578 (119867 (1205721) 119891))

1015840

minus (1 minus 120582) 120578 (119861120578 (119867 (1205721) 119891))

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840= minus119887119901119892 (119911) + 119887

2minus 119887 + 120578

(29)

Now taking logarithmic derivative we have

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)10158401015840+ 120582 (119867 (1205721) 119891)

1015840+ (1 minus 120582) (119867 (1205721) 119891)

1015840]

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]

minus

(1 minus 120582) 119861120578 (119867 (1205721) 119891)1015840+ 120582119911 (119861120578 (119867 (1205721) 119891))

10158401015840

+ 120582 (119861120578 (119867 (1205721) 119891))1015840

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

=minus1198871199011198921015840(119911)

minus119887119901119892 (119911) + 1198872 minus 119887

minus1

119901[1 +

1

119887(119911 (119867 (1205721) 119891)

1015840+ 1205821199112(119867 (1205721) 119891)

10158401015840

(1 minus 120582) (119867 (1205721) 119891) + 120582119911 (119867 (1205721) 119891)1015840) minus 119887] = 119892 (119911) +

1199111198921015840(119911)

minus119887119901119892 (119911) + 1198872 minus 119887

(30)

Using Lemma 2 we get 119892(119911) ≺ 119902119896120574(119911) which implies

119861120578 (119891 (119911)) isin 119872119876119901 (119887 119896 120582 1205721) (31)

This proves the assertion

Now we get coefficient estimates of the class 119872119876119901(119887 119896120582 1205721)

Theorem 6 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) and 119891(119911) is given by(1) then

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 1198860 = 1 (32)

provided2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

The Scientific World Journal 5

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(33)

for all 119896 le 119899

Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave

minus1

119901[1

+1

119887(119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= 119887ℎ (119911) +119887

119901minus1198872

119901

(34)

which gives

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901

= [119887ℎ (119911) +119887 (1 minus 119887)

119901]

sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]

(35)

Let us write119867(1205721)119891(119911) = 119891(119911) then

119891 (119911) = 119911minus119901+

infin

sum

119899=0

119886119899119911119899minus119901

1198911015840(119911) = minus119901119911

minus119901minus1+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901minus1

1199111198911015840(119911) = minus119901119911

minus119901+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901

11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901minus2

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2

119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901

(36)

Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes

119911minus119901minus

infin

sum

119899=0

119886119899

(119899 minus 119901)

119901119911119899minus119901+ 120582(minus (119901 + 1) 119911

minus119901

+

infin

sum

119899=0

119886119899

(119899 minus 119901) (119899 minus 119901 minus 1)

119901119911119899minus119901)

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887

+

infin

sum

119899=1

119887119888119899119911119899+119887 (1 minus 119887)

119901] [(1 minus 120582119901 minus 120582) 119911

minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(37)

From (37) we have

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887 + 1198871198881119911

+ 11988711988821199112+ 1198871198883119911

3+ sdot sdot sdot + 119887119888119899119911

119899+ sdot sdot sdot +

119887 (1 minus 119887)

119901]

sdot [(1 minus 120582119901 minus 120582) 119911minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(38)

Now comparing coefficients of 1199111minus119901 we have

minus 1198861

1 minus 119901

119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (

119887 (1 minus 119887)

119901)

sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)

minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1

119901]

= 1198871198881 (1 minus 120582119901 minus 120582)

1198861 = minus119901

(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)

sdot 1198871198881

(39)

6 The Scientific World Journal

and comparing the coefficients of 1199112minus119901 gives

minus 1198862 [2 minus 119901

119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887

+119887 (1 minus 119887)

119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901

119901+119887119901 + 119887 (1 minus 119887)

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

1198862 = minus119901

(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1

minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)

(40)

and for the coefficient of 1199113minus119901 we have

minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3

119901]

= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1

minus 120582119901 minus 120582]

1198863 = minus119901

(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1

minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]

sdot 11988711988811198862)

(41)

which generalise to

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1

minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)

sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]

(42)

The above expression can also be written as

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)

sdot

119899

sum

119896=1

[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1

with 1198860 = 1

(43)

Now taking

119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582

1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)

we have1003816100381610038161003816119860119899119896

1003816100381610038161003816 lt 1

if

2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(45)

for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have

10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)

where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give

100381610038161003816100381611988611003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 1

10038161003816100381610038161003816100381610038161003816

100381610038161003816100381611988621003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 2

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816)

100381610038161003816100381611988631003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 3

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +

1003816100381610038161003816119886119899minus11003816100381610038161003816)

(47)

This can also be written as

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 (48)

This concludes the proof

Competing Interests

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

Acknowledgments

The work here is supported by AP-2013-009

References

[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000

[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004

[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015

[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983

The Scientific World Journal 7

[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945

[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008

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 2: MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World Journal andcomparingthecoe cientsof 2 gives 2 s 2 ti1+T 1 j= 2 i1+T 1 jsV V (1

2 The Scientific World Journal

The generalised hypergeometric function119902119865119904(1205721 120572119902

1205731 120573119904 119911) for complex parameters1205721 120572119902 and1205731 120573119904with 120573119895 = 0 minus1 minus2 minus3 for 119895 = 1 2 3 119904 is defined as

119902119865119904 (1205721 120572119902 1205731 120573119904 119911) =

infin

sum

119899=0

(1205721)119899sdot sdot sdot (120572119902)119899

(1205731)119899sdot sdot sdot (120573119904)119899

119899119911119899

= 1 +

infin

sum

119899=1

(1205721)119899sdot sdot sdot (120572119902)119899

(1205731)119899sdot sdot sdot (120573119904)119899

119911119899

(119899)

(5)

with 119902 le 119904+1 119902 119904 isin N0 = Ncup0 and (120572)119899 is the well-knownPochhammer symbol related to the factorial and the Gammafunction by the relation

(120572)119899 =(120572 + 119899 minus 1)

(120572 minus 1)=Γ (120572 + 119899)

Γ (120572) (6)

Also (5) implies

ℎ119901 (1205721 120572119902 1205731 120573119904 119911)

= 119911minus119901

119902119865119904 (1205721 120572119902 1205731 120573119904 119911) isin 119872119901

(7)

Liu and Srivastava [2] defined a linear operator for func-tions belonging to the class of multivalent meromorphicfunction119867119901(1205721 120572119902 1205731 120573119904) 119872119901 rarr119872119901 as follows

119867119901 (1205721 120572119902 1205731 120573119904)

= ℎ119901 (1205721 120572119902 1205731 120573119904 119911) ⋆ 119891 (119911)

(8)

If we assume for brevity that 119867119901119902119904(1205721) = 119867119901(1205721 120572119902

1205731 120573119904) then the following identity holds for this operator

119911 [119867119901119902119904 (1205721) 119891 (119911)]1015840

= 1205721119867119901119902119904 (1205721 + 1) 119891 (119911)

minus (1205721 + 119901)119867119901119902119904 (1205721) 119891 (119911)

(9)

Shareef [3] defined and studied subclass119872119876119901(119896 120582 1205721) ofmeromorphic function associated with conic domain for 119896 ge0 0 le 120582 lt 1 and 119901 ge 1 as follows

minus1

119901

[

[

119911 (119867119901119902119904 (1205721) 119891 (119911))1015840

+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840]

]

≺ 119902119896120574 (119911)

(10)

We now define a new subclass119872119876119901(119887 119896 120582 1205721) of mero-morphic function associated with conic domain for 119896 ge 00 le 120582 lt 1 119901 ge 1 and 119887 ge 1 as follows

minus1

119901

[

[

1

+1

119887(

119911 (119867119901119902119904 (1205721) 119891 (119911))1015840

+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840)

minus 119887]

]

≺ 119902119896120574 (119911)

(11)

Since 119902119896120574 is a convex and univalent function for ℎ(119911) ≺119902119896120574(119911) it means ℎ(119864lowast) is contained in 119902119896120574(119864

lowast) where

ℎ (119911) = minus1

119901

[

[

1

+1

119887(

119911 (119867119901119902119904 (1205721) 119891 (119911))1015840

+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840)

minus 119887]

]

(12)

In the next two sections for brevity we drop the subscripts ofthe operator119867119901119902119904(1205721)

2 Preliminary Results

Lemma 1 (see [4]) Let ℎ2(119911) be convex in 119864 andR(120582ℎ2(119911) +120583) gt 0 where 120583 isin C 120582 isin C0 and 119911 isin 119864 If ℎ1(119911) is analyticin 119864 with ℎ1(0) = ℎ2(0) then

ℎ1 (119911) +119911ℎ1015840

1(119911)

120582ℎ1 (119911) + 120583≺ ℎ2 (119911)

119894119898119901119897119894119890119904 ℎ1 (119911) ≺ ℎ2 (119911)

(13)

Lemma 2 (see [5]) Let ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 and 119867(119911) =

1 + suminfin

119899=1119889119899119911119899 and ℎ ≺ 119867 If 119867(119911) is univalent and convex in

119864 then10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611988911003816100381610038161003816 (14)

for 119899 ge 1

Lemma 3 (see [6]) If 119902119896120574(119911) = 1 + 1199021119911 + 11990221199112 + sdot sdot sdot then

1199021 =

8 (1 minus 120574) (arccos (119896))2

1205872 (1 minus 1198962) 0 lt 119896 lt 1

8 (1 minus 120574)

1205872 119896 = 1

1205872(1 minus 120574)

4radic119905 (1198962 minus 1) 1198962 (119905) (1 + 119905) 119896 gt 1

(15)

One now states and proves the main results

3 Main Results

In this section we explore some of the geometric propertiesexhibited by the class119872119876119901(119887 119896 120582 1205721)

We begin by discussing an inclusion property for the class119872119876119901(119887 119896 120582 1205721)

Theorem 4 IfR(1205721) gt 119901R(119887119902119896120582(119911) minus 1) then

119872119876119901 (119887 119896 120582 1205721 + 1) sub 119872119876119901 (119887 119896 120582 1205721) (16)

The Scientific World Journal 3

Proof Let 119891 isin 119872119876119901(119887 119896 120582 1205721 + 1) and set

minus1

119901[1

+1

119887(119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840)

minus 119887] = ℎ (119911)

(17)

But differentiating (9) with respect to 119911 we get

[119911119867 (1205721) 119891 (119911)]1015840

= 1205721119867((1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840

119911 [119867 (1205721) 119891 (119911)]10158401015840+ [119867 (1205721) 119891 (119911)]

1015840

= 1205721119867((1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840

1199112(119867 (1205721) 119891 (119911))

10158401015840

= 1205721119911 (119867 (1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901 + 1) 119911 (119867 (1205721) 119891 (119911))1015840

(18)

Putting (18) in (17) we have

1205821199111205721 (119867 (120572 + 1) 119891 (119911))1015840+ [1 minus 120582 (1205721 + 119901 + 1)] 119911 (119867 (1205721) 119891 (119911))

1015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= minus119887119901ℎ (119911)

1205821205721119911 (119867 (1205721 + 1) 119891 (119911))1015840+ (1 minus 120582) 1205721 (119867 (1205721 + 1) 119891 (119911))

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= minus119887119901ℎ (119911) + (1205721 + 119901)

(19)

Taking logarithmic derivative of (19) we have

119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840

+ 119887119901ℎ (119911) + 119887 (119887 minus 1) =minus119887119901119911ℎ

1015840(119911)

minus119887119901ℎ (119911) + 1205721 + 119901

119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840

= minus119887119901[ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + 1205721 + 119901] minus 119887 (119887 minus 1)

minus1

119901[1

+1

119887(119911 (119867 (1205721 + 1) 119891 (119911))

1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721 + 1) 119891 (119911) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840)

minus 119887] = ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + (1205721 + 119901)

(20)

Since 119891 isin 119872119876119902(119887 119896 120582 1205721 + 1) therefore

ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + (1205721 + 119901)≺ 119902119896120574 (119911) (21)

Using Lemma 2 we have

ℎ (119911) ≺ 119902119896120574 (119911) (22)

providedR(minus119887119901119902119896120582(119911)+1205721 +119901) gt 0 or equivalentlyR(1205721) gt119901R(119887119902119896120574(119911) minus 1) Hence 119891 isin 119872119876119901(119887 119896 120582 1205721)

We now show that the class 119872119876119901(119887 119896 120582 1205721) is closedunder a certain integral

Theorem 5 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then the integral

119861120578 (119867 (120572) 119891 (119911)) =120578 minus 119901

119911120578int

119911

0

119905120578minus1(119867 (120572) 119891 (119911)) d119905 (23)

maps 119891(119911) into119872119876119901(119887 119896 120582 1205721)

Proof From (23) we have

119911120578119861120578 (119867 (120572) 119891 (119911)) = (120578 minus 119901)int

119911

0

119905120578minus1(119867 (120572) 119891 (119911)) d119905 (24)

Note that

119861120578 (119867 (120572) 119891 (119911)) = (119867 (120572)) 119861120578119891 (119911) (25)

Differentiating (24) above we get

120578119911120578minus1119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))

1015840

= (120578 minus 119901) 119911120578minus1(119867 (120572) 119891 (119911))

4 The Scientific World Journal

120578119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))1015840

= (120578 minus 119901) (119867 (120572) 119891 (119911))

(26)

Differentiate again

120578 (119861120578 (119867 (120572) 119891 (119911)))1015840

+ 119911 (119861120578 (119867 (120572) 119891 (119911)))10158401015840

+ (119861120578 (119867 (120572) 119891 (119911)))1015840

= (120578 minus 119901) (119867 (120572) 119891 (119911))1015840

119911 (119861120578 (119891 (119911)))10158401015840

= (120578 minus 119901) (119861120578 (119867 (120572) 119891 (119911)))1015840

minus (120578 + 1) (119861120578 (119891 (119911)))1015840

(27)

Now let

minus1

119901

[

[

1

+1

119887(

119911 (119861120578 (119867 (1205721) 119891))1015840

+ 1205821199112(119861120578 (119867 (1205721) 119891))

10158401015840

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840)

minus 119887]

]

= 119892 (119911)

(28)

Using (26) and (27) in (28) we get

119911119861120578 (119867 (1205721) 119891)1015840+ 120582 [119911 (120578 minus 119901) (119867 (1205721) 119891)

1015840minus 119911 (1 + 120578) (119861120578 (119867 (1205721) 119891))

1015840

]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

120582119911 (120578 minus 119901) (119867 (1205721) 119891)1015840+ [1 minus 120582 (1 + 120578)] [(120578 minus 119901) (119867 (1205721) 119891) minus 120578 (119861120578 (119867 (1205721) 119891))]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)] minus 120582120578119911 (119861120578 (119867 (1205721) 119891))

1015840

minus (1 minus 120582) 120578 (119861120578 (119867 (1205721) 119891))

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840= minus119887119901119892 (119911) + 119887

2minus 119887 + 120578

(29)

Now taking logarithmic derivative we have

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)10158401015840+ 120582 (119867 (1205721) 119891)

1015840+ (1 minus 120582) (119867 (1205721) 119891)

1015840]

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]

minus

(1 minus 120582) 119861120578 (119867 (1205721) 119891)1015840+ 120582119911 (119861120578 (119867 (1205721) 119891))

10158401015840

+ 120582 (119861120578 (119867 (1205721) 119891))1015840

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

=minus1198871199011198921015840(119911)

minus119887119901119892 (119911) + 1198872 minus 119887

minus1

119901[1 +

1

119887(119911 (119867 (1205721) 119891)

1015840+ 1205821199112(119867 (1205721) 119891)

10158401015840

(1 minus 120582) (119867 (1205721) 119891) + 120582119911 (119867 (1205721) 119891)1015840) minus 119887] = 119892 (119911) +

1199111198921015840(119911)

minus119887119901119892 (119911) + 1198872 minus 119887

(30)

Using Lemma 2 we get 119892(119911) ≺ 119902119896120574(119911) which implies

119861120578 (119891 (119911)) isin 119872119876119901 (119887 119896 120582 1205721) (31)

This proves the assertion

Now we get coefficient estimates of the class 119872119876119901(119887 119896120582 1205721)

Theorem 6 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) and 119891(119911) is given by(1) then

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 1198860 = 1 (32)

provided2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

The Scientific World Journal 5

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(33)

for all 119896 le 119899

Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave

minus1

119901[1

+1

119887(119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= 119887ℎ (119911) +119887

119901minus1198872

119901

(34)

which gives

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901

= [119887ℎ (119911) +119887 (1 minus 119887)

119901]

sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]

(35)

Let us write119867(1205721)119891(119911) = 119891(119911) then

119891 (119911) = 119911minus119901+

infin

sum

119899=0

119886119899119911119899minus119901

1198911015840(119911) = minus119901119911

minus119901minus1+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901minus1

1199111198911015840(119911) = minus119901119911

minus119901+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901

11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901minus2

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2

119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901

(36)

Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes

119911minus119901minus

infin

sum

119899=0

119886119899

(119899 minus 119901)

119901119911119899minus119901+ 120582(minus (119901 + 1) 119911

minus119901

+

infin

sum

119899=0

119886119899

(119899 minus 119901) (119899 minus 119901 minus 1)

119901119911119899minus119901)

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887

+

infin

sum

119899=1

119887119888119899119911119899+119887 (1 minus 119887)

119901] [(1 minus 120582119901 minus 120582) 119911

minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(37)

From (37) we have

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887 + 1198871198881119911

+ 11988711988821199112+ 1198871198883119911

3+ sdot sdot sdot + 119887119888119899119911

119899+ sdot sdot sdot +

119887 (1 minus 119887)

119901]

sdot [(1 minus 120582119901 minus 120582) 119911minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(38)

Now comparing coefficients of 1199111minus119901 we have

minus 1198861

1 minus 119901

119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (

119887 (1 minus 119887)

119901)

sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)

minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1

119901]

= 1198871198881 (1 minus 120582119901 minus 120582)

1198861 = minus119901

(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)

sdot 1198871198881

(39)

6 The Scientific World Journal

and comparing the coefficients of 1199112minus119901 gives

minus 1198862 [2 minus 119901

119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887

+119887 (1 minus 119887)

119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901

119901+119887119901 + 119887 (1 minus 119887)

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

1198862 = minus119901

(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1

minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)

(40)

and for the coefficient of 1199113minus119901 we have

minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3

119901]

= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1

minus 120582119901 minus 120582]

1198863 = minus119901

(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1

minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]

sdot 11988711988811198862)

(41)

which generalise to

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1

minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)

sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]

(42)

The above expression can also be written as

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)

sdot

119899

sum

119896=1

[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1

with 1198860 = 1

(43)

Now taking

119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582

1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)

we have1003816100381610038161003816119860119899119896

1003816100381610038161003816 lt 1

if

2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(45)

for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have

10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)

where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give

100381610038161003816100381611988611003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 1

10038161003816100381610038161003816100381610038161003816

100381610038161003816100381611988621003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 2

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816)

100381610038161003816100381611988631003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 3

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +

1003816100381610038161003816119886119899minus11003816100381610038161003816)

(47)

This can also be written as

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 (48)

This concludes the proof

Competing Interests

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

Acknowledgments

The work here is supported by AP-2013-009

References

[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000

[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004

[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015

[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983

The Scientific World Journal 7

[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945

[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008

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 3: MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World Journal andcomparingthecoe cientsof 2 gives 2 s 2 ti1+T 1 j= 2 i1+T 1 jsV V (1

The Scientific World Journal 3

Proof Let 119891 isin 119872119876119901(119887 119896 120582 1205721 + 1) and set

minus1

119901[1

+1

119887(119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840)

minus 119887] = ℎ (119911)

(17)

But differentiating (9) with respect to 119911 we get

[119911119867 (1205721) 119891 (119911)]1015840

= 1205721119867((1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840

119911 [119867 (1205721) 119891 (119911)]10158401015840+ [119867 (1205721) 119891 (119911)]

1015840

= 1205721119867((1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840

1199112(119867 (1205721) 119891 (119911))

10158401015840

= 1205721119911 (119867 (1205721 + 1) 119891 (119911))1015840

minus (1205721 + 119901 + 1) 119911 (119867 (1205721) 119891 (119911))1015840

(18)

Putting (18) in (17) we have

1205821199111205721 (119867 (120572 + 1) 119891 (119911))1015840+ [1 minus 120582 (1205721 + 119901 + 1)] 119911 (119867 (1205721) 119891 (119911))

1015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= minus119887119901ℎ (119911)

1205821205721119911 (119867 (1205721 + 1) 119891 (119911))1015840+ (1 minus 120582) 1205721 (119867 (1205721 + 1) 119891 (119911))

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= minus119887119901ℎ (119911) + (1205721 + 119901)

(19)

Taking logarithmic derivative of (19) we have

119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840

+ 119887119901ℎ (119911) + 119887 (119887 minus 1) =minus119887119901119911ℎ

1015840(119911)

minus119887119901ℎ (119911) + 1205721 + 119901

119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840

= minus119887119901[ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + 1205721 + 119901] minus 119887 (119887 minus 1)

minus1

119901[1

+1

119887(119911 (119867 (1205721 + 1) 119891 (119911))

1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721 + 1) 119891 (119911) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840)

minus 119887] = ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + (1205721 + 119901)

(20)

Since 119891 isin 119872119876119902(119887 119896 120582 1205721 + 1) therefore

ℎ (119911) +119911ℎ1015840(119911)

minus119887119901ℎ (119911) + (1205721 + 119901)≺ 119902119896120574 (119911) (21)

Using Lemma 2 we have

ℎ (119911) ≺ 119902119896120574 (119911) (22)

providedR(minus119887119901119902119896120582(119911)+1205721 +119901) gt 0 or equivalentlyR(1205721) gt119901R(119887119902119896120574(119911) minus 1) Hence 119891 isin 119872119876119901(119887 119896 120582 1205721)

We now show that the class 119872119876119901(119887 119896 120582 1205721) is closedunder a certain integral

Theorem 5 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then the integral

119861120578 (119867 (120572) 119891 (119911)) =120578 minus 119901

119911120578int

119911

0

119905120578minus1(119867 (120572) 119891 (119911)) d119905 (23)

maps 119891(119911) into119872119876119901(119887 119896 120582 1205721)

Proof From (23) we have

119911120578119861120578 (119867 (120572) 119891 (119911)) = (120578 minus 119901)int

119911

0

119905120578minus1(119867 (120572) 119891 (119911)) d119905 (24)

Note that

119861120578 (119867 (120572) 119891 (119911)) = (119867 (120572)) 119861120578119891 (119911) (25)

Differentiating (24) above we get

120578119911120578minus1119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))

1015840

= (120578 minus 119901) 119911120578minus1(119867 (120572) 119891 (119911))

4 The Scientific World Journal

120578119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))1015840

= (120578 minus 119901) (119867 (120572) 119891 (119911))

(26)

Differentiate again

120578 (119861120578 (119867 (120572) 119891 (119911)))1015840

+ 119911 (119861120578 (119867 (120572) 119891 (119911)))10158401015840

+ (119861120578 (119867 (120572) 119891 (119911)))1015840

= (120578 minus 119901) (119867 (120572) 119891 (119911))1015840

119911 (119861120578 (119891 (119911)))10158401015840

= (120578 minus 119901) (119861120578 (119867 (120572) 119891 (119911)))1015840

minus (120578 + 1) (119861120578 (119891 (119911)))1015840

(27)

Now let

minus1

119901

[

[

1

+1

119887(

119911 (119861120578 (119867 (1205721) 119891))1015840

+ 1205821199112(119861120578 (119867 (1205721) 119891))

10158401015840

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840)

minus 119887]

]

= 119892 (119911)

(28)

Using (26) and (27) in (28) we get

119911119861120578 (119867 (1205721) 119891)1015840+ 120582 [119911 (120578 minus 119901) (119867 (1205721) 119891)

1015840minus 119911 (1 + 120578) (119861120578 (119867 (1205721) 119891))

1015840

]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

120582119911 (120578 minus 119901) (119867 (1205721) 119891)1015840+ [1 minus 120582 (1 + 120578)] [(120578 minus 119901) (119867 (1205721) 119891) minus 120578 (119861120578 (119867 (1205721) 119891))]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)] minus 120582120578119911 (119861120578 (119867 (1205721) 119891))

1015840

minus (1 minus 120582) 120578 (119861120578 (119867 (1205721) 119891))

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840= minus119887119901119892 (119911) + 119887

2minus 119887 + 120578

(29)

Now taking logarithmic derivative we have

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)10158401015840+ 120582 (119867 (1205721) 119891)

1015840+ (1 minus 120582) (119867 (1205721) 119891)

1015840]

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]

minus

(1 minus 120582) 119861120578 (119867 (1205721) 119891)1015840+ 120582119911 (119861120578 (119867 (1205721) 119891))

10158401015840

+ 120582 (119861120578 (119867 (1205721) 119891))1015840

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

=minus1198871199011198921015840(119911)

minus119887119901119892 (119911) + 1198872 minus 119887

minus1

119901[1 +

1

119887(119911 (119867 (1205721) 119891)

1015840+ 1205821199112(119867 (1205721) 119891)

10158401015840

(1 minus 120582) (119867 (1205721) 119891) + 120582119911 (119867 (1205721) 119891)1015840) minus 119887] = 119892 (119911) +

1199111198921015840(119911)

minus119887119901119892 (119911) + 1198872 minus 119887

(30)

Using Lemma 2 we get 119892(119911) ≺ 119902119896120574(119911) which implies

119861120578 (119891 (119911)) isin 119872119876119901 (119887 119896 120582 1205721) (31)

This proves the assertion

Now we get coefficient estimates of the class 119872119876119901(119887 119896120582 1205721)

Theorem 6 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) and 119891(119911) is given by(1) then

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 1198860 = 1 (32)

provided2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

The Scientific World Journal 5

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(33)

for all 119896 le 119899

Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave

minus1

119901[1

+1

119887(119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= 119887ℎ (119911) +119887

119901minus1198872

119901

(34)

which gives

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901

= [119887ℎ (119911) +119887 (1 minus 119887)

119901]

sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]

(35)

Let us write119867(1205721)119891(119911) = 119891(119911) then

119891 (119911) = 119911minus119901+

infin

sum

119899=0

119886119899119911119899minus119901

1198911015840(119911) = minus119901119911

minus119901minus1+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901minus1

1199111198911015840(119911) = minus119901119911

minus119901+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901

11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901minus2

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2

119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901

(36)

Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes

119911minus119901minus

infin

sum

119899=0

119886119899

(119899 minus 119901)

119901119911119899minus119901+ 120582(minus (119901 + 1) 119911

minus119901

+

infin

sum

119899=0

119886119899

(119899 minus 119901) (119899 minus 119901 minus 1)

119901119911119899minus119901)

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887

+

infin

sum

119899=1

119887119888119899119911119899+119887 (1 minus 119887)

119901] [(1 minus 120582119901 minus 120582) 119911

minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(37)

From (37) we have

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887 + 1198871198881119911

+ 11988711988821199112+ 1198871198883119911

3+ sdot sdot sdot + 119887119888119899119911

119899+ sdot sdot sdot +

119887 (1 minus 119887)

119901]

sdot [(1 minus 120582119901 minus 120582) 119911minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(38)

Now comparing coefficients of 1199111minus119901 we have

minus 1198861

1 minus 119901

119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (

119887 (1 minus 119887)

119901)

sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)

minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1

119901]

= 1198871198881 (1 minus 120582119901 minus 120582)

1198861 = minus119901

(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)

sdot 1198871198881

(39)

6 The Scientific World Journal

and comparing the coefficients of 1199112minus119901 gives

minus 1198862 [2 minus 119901

119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887

+119887 (1 minus 119887)

119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901

119901+119887119901 + 119887 (1 minus 119887)

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

1198862 = minus119901

(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1

minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)

(40)

and for the coefficient of 1199113minus119901 we have

minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3

119901]

= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1

minus 120582119901 minus 120582]

1198863 = minus119901

(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1

minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]

sdot 11988711988811198862)

(41)

which generalise to

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1

minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)

sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]

(42)

The above expression can also be written as

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)

sdot

119899

sum

119896=1

[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1

with 1198860 = 1

(43)

Now taking

119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582

1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)

we have1003816100381610038161003816119860119899119896

1003816100381610038161003816 lt 1

if

2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(45)

for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have

10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)

where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give

100381610038161003816100381611988611003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 1

10038161003816100381610038161003816100381610038161003816

100381610038161003816100381611988621003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 2

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816)

100381610038161003816100381611988631003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 3

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +

1003816100381610038161003816119886119899minus11003816100381610038161003816)

(47)

This can also be written as

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 (48)

This concludes the proof

Competing Interests

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

Acknowledgments

The work here is supported by AP-2013-009

References

[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000

[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004

[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015

[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983

The Scientific World Journal 7

[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945

[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008

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 4: MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World Journal andcomparingthecoe cientsof 2 gives 2 s 2 ti1+T 1 j= 2 i1+T 1 jsV V (1

4 The Scientific World Journal

120578119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))1015840

= (120578 minus 119901) (119867 (120572) 119891 (119911))

(26)

Differentiate again

120578 (119861120578 (119867 (120572) 119891 (119911)))1015840

+ 119911 (119861120578 (119867 (120572) 119891 (119911)))10158401015840

+ (119861120578 (119867 (120572) 119891 (119911)))1015840

= (120578 minus 119901) (119867 (120572) 119891 (119911))1015840

119911 (119861120578 (119891 (119911)))10158401015840

= (120578 minus 119901) (119861120578 (119867 (120572) 119891 (119911)))1015840

minus (120578 + 1) (119861120578 (119891 (119911)))1015840

(27)

Now let

minus1

119901

[

[

1

+1

119887(

119911 (119861120578 (119867 (1205721) 119891))1015840

+ 1205821199112(119861120578 (119867 (1205721) 119891))

10158401015840

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840)

minus 119887]

]

= 119892 (119911)

(28)

Using (26) and (27) in (28) we get

119911119861120578 (119867 (1205721) 119891)1015840+ 120582 [119911 (120578 minus 119901) (119867 (1205721) 119891)

1015840minus 119911 (1 + 120578) (119861120578 (119867 (1205721) 119891))

1015840

]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

120582119911 (120578 minus 119901) (119867 (1205721) 119891)1015840+ [1 minus 120582 (1 + 120578)] [(120578 minus 119901) (119867 (1205721) 119891) minus 120578 (119861120578 (119867 (1205721) 119891))]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)] minus 120582120578119911 (119861120578 (119867 (1205721) 119891))

1015840

minus (1 minus 120582) 120578 (119861120578 (119867 (1205721) 119891))

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

= minus119887119901119892 (119911) + 1198872minus 119887

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840= minus119887119901119892 (119911) + 119887

2minus 119887 + 120578

(29)

Now taking logarithmic derivative we have

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)10158401015840+ 120582 (119867 (1205721) 119891)

1015840+ (1 minus 120582) (119867 (1205721) 119891)

1015840]

(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]

minus

(1 minus 120582) 119861120578 (119867 (1205721) 119891)1015840+ 120582119911 (119861120578 (119867 (1205721) 119891))

10158401015840

+ 120582 (119861120578 (119867 (1205721) 119891))1015840

(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840

=minus1198871199011198921015840(119911)

minus119887119901119892 (119911) + 1198872 minus 119887

minus1

119901[1 +

1

119887(119911 (119867 (1205721) 119891)

1015840+ 1205821199112(119867 (1205721) 119891)

10158401015840

(1 minus 120582) (119867 (1205721) 119891) + 120582119911 (119867 (1205721) 119891)1015840) minus 119887] = 119892 (119911) +

1199111198921015840(119911)

minus119887119901119892 (119911) + 1198872 minus 119887

(30)

Using Lemma 2 we get 119892(119911) ≺ 119902119896120574(119911) which implies

119861120578 (119891 (119911)) isin 119872119876119901 (119887 119896 120582 1205721) (31)

This proves the assertion

Now we get coefficient estimates of the class 119872119876119901(119887 119896120582 1205721)

Theorem 6 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) and 119891(119911) is given by(1) then

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 1198860 = 1 (32)

provided2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

The Scientific World Journal 5

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(33)

for all 119896 le 119899

Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave

minus1

119901[1

+1

119887(119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= 119887ℎ (119911) +119887

119901minus1198872

119901

(34)

which gives

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901

= [119887ℎ (119911) +119887 (1 minus 119887)

119901]

sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]

(35)

Let us write119867(1205721)119891(119911) = 119891(119911) then

119891 (119911) = 119911minus119901+

infin

sum

119899=0

119886119899119911119899minus119901

1198911015840(119911) = minus119901119911

minus119901minus1+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901minus1

1199111198911015840(119911) = minus119901119911

minus119901+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901

11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901minus2

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2

119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901

(36)

Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes

119911minus119901minus

infin

sum

119899=0

119886119899

(119899 minus 119901)

119901119911119899minus119901+ 120582(minus (119901 + 1) 119911

minus119901

+

infin

sum

119899=0

119886119899

(119899 minus 119901) (119899 minus 119901 minus 1)

119901119911119899minus119901)

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887

+

infin

sum

119899=1

119887119888119899119911119899+119887 (1 minus 119887)

119901] [(1 minus 120582119901 minus 120582) 119911

minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(37)

From (37) we have

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887 + 1198871198881119911

+ 11988711988821199112+ 1198871198883119911

3+ sdot sdot sdot + 119887119888119899119911

119899+ sdot sdot sdot +

119887 (1 minus 119887)

119901]

sdot [(1 minus 120582119901 minus 120582) 119911minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(38)

Now comparing coefficients of 1199111minus119901 we have

minus 1198861

1 minus 119901

119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (

119887 (1 minus 119887)

119901)

sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)

minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1

119901]

= 1198871198881 (1 minus 120582119901 minus 120582)

1198861 = minus119901

(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)

sdot 1198871198881

(39)

6 The Scientific World Journal

and comparing the coefficients of 1199112minus119901 gives

minus 1198862 [2 minus 119901

119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887

+119887 (1 minus 119887)

119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901

119901+119887119901 + 119887 (1 minus 119887)

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

1198862 = minus119901

(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1

minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)

(40)

and for the coefficient of 1199113minus119901 we have

minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3

119901]

= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1

minus 120582119901 minus 120582]

1198863 = minus119901

(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1

minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]

sdot 11988711988811198862)

(41)

which generalise to

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1

minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)

sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]

(42)

The above expression can also be written as

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)

sdot

119899

sum

119896=1

[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1

with 1198860 = 1

(43)

Now taking

119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582

1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)

we have1003816100381610038161003816119860119899119896

1003816100381610038161003816 lt 1

if

2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(45)

for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have

10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)

where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give

100381610038161003816100381611988611003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 1

10038161003816100381610038161003816100381610038161003816

100381610038161003816100381611988621003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 2

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816)

100381610038161003816100381611988631003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 3

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +

1003816100381610038161003816119886119899minus11003816100381610038161003816)

(47)

This can also be written as

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 (48)

This concludes the proof

Competing Interests

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

Acknowledgments

The work here is supported by AP-2013-009

References

[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000

[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004

[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015

[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983

The Scientific World Journal 7

[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945

[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008

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 5: MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World Journal andcomparingthecoe cientsof 2 gives 2 s 2 ti1+T 1 j= 2 i1+T 1 jsV V (1

The Scientific World Journal 5

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582

119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(33)

for all 119896 le 119899

Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave

minus1

119901[1

+1

119887(119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840

= 119887ℎ (119911) +119887

119901minus1198872

119901

(34)

which gives

minus119911 (119867 (1205721) 119891 (119911))

1015840+ 1205821199112(119867 (1205721) 119891 (119911))

10158401015840

119901

= [119887ℎ (119911) +119887 (1 minus 119887)

119901]

sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]

(35)

Let us write119867(1205721)119891(119911) = 119891(119911) then

119891 (119911) = 119911minus119901+

infin

sum

119899=0

119886119899119911119899minus119901

1198911015840(119911) = minus119901119911

minus119901minus1+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901minus1

1199111198911015840(119911) = minus119901119911

minus119901+

infin

sum

119899=0

119886119899 (119899 minus 119901) 119911119899minus119901

11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901minus2

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2

119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911

minus119901

+

infin

sum

119899=0

119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901

(36)

Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes

119911minus119901minus

infin

sum

119899=0

119886119899

(119899 minus 119901)

119901119911119899minus119901+ 120582(minus (119901 + 1) 119911

minus119901

+

infin

sum

119899=0

119886119899

(119899 minus 119901) (119899 minus 119901 minus 1)

119901119911119899minus119901)

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887

+

infin

sum

119899=1

119887119888119899119911119899+119887 (1 minus 119887)

119901] [(1 minus 120582119901 minus 120582) 119911

minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(37)

From (37) we have

(1 minus 120582119901 minus 120582) 119911minus119901

minus

infin

sum

119899=1

119886119899

119899 minus 119901

119901[1 + 120582 (119899 minus 119901 minus 1)] 119911

119899minus119901= [119887 + 1198871198881119911

+ 11988711988821199112+ 1198871198883119911

3+ sdot sdot sdot + 119887119888119899119911

119899+ sdot sdot sdot +

119887 (1 minus 119887)

119901]

sdot [(1 minus 120582119901 minus 120582) 119911minus119901

+

infin

sum

119899=1

119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]

(38)

Now comparing coefficients of 1199111minus119901 we have

minus 1198861

1 minus 119901

119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (

119887 (1 minus 119887)

119901)

sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)

minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1

119901]

= 1198871198881 (1 minus 120582119901 minus 120582)

1198861 = minus119901

(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)

sdot 1198871198881

(39)

6 The Scientific World Journal

and comparing the coefficients of 1199112minus119901 gives

minus 1198862 [2 minus 119901

119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887

+119887 (1 minus 119887)

119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901

119901+119887119901 + 119887 (1 minus 119887)

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

1198862 = minus119901

(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1

minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)

(40)

and for the coefficient of 1199113minus119901 we have

minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3

119901]

= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1

minus 120582119901 minus 120582]

1198863 = minus119901

(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1

minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]

sdot 11988711988811198862)

(41)

which generalise to

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1

minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)

sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]

(42)

The above expression can also be written as

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)

sdot

119899

sum

119896=1

[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1

with 1198860 = 1

(43)

Now taking

119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582

1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)

we have1003816100381610038161003816119860119899119896

1003816100381610038161003816 lt 1

if

2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(45)

for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have

10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)

where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give

100381610038161003816100381611988611003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 1

10038161003816100381610038161003816100381610038161003816

100381610038161003816100381611988621003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 2

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816)

100381610038161003816100381611988631003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 3

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +

1003816100381610038161003816119886119899minus11003816100381610038161003816)

(47)

This can also be written as

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 (48)

This concludes the proof

Competing Interests

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

Acknowledgments

The work here is supported by AP-2013-009

References

[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000

[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004

[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015

[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983

The Scientific World Journal 7

[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945

[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008

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 6: MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World Journal andcomparingthecoe cientsof 2 gives 2 s 2 ti1+T 1 j= 2 i1+T 1 jsV V (1

6 The Scientific World Journal

and comparing the coefficients of 1199112minus119901 gives

minus 1198862 [2 minus 119901

119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887

+119887 (1 minus 119887)

119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901

119901+119887119901 + 119887 (1 minus 119887)

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2

119901] = 1198861 [1

minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)

1198862 = minus119901

(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1

minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)

(40)

and for the coefficient of 1199113minus119901 we have

minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3

119901]

= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1

minus 120582119901 minus 120582]

1198863 = minus119901

(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1

minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]

sdot 11988711988811198862)

(41)

which generalise to

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1

minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)

sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]

(42)

The above expression can also be written as

119886119899 = minus119901

(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)

sdot

119899

sum

119896=1

[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1

with 1198860 = 1

(43)

Now taking

119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582

1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)

we have1003816100381610038161003816119860119899119896

1003816100381610038161003816 lt 1

if

2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0

2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0

(45)

for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have

10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)

where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give

100381610038161003816100381611988611003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 1

10038161003816100381610038161003816100381610038161003816

100381610038161003816100381611988621003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 2

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816)

100381610038161003816100381611988631003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 3

10038161003816100381610038161003816100381610038161003816

(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +

1003816100381610038161003816119886119899minus11003816100381610038161003816)

(47)

This can also be written as

10038161003816100381610038161198861198991003816100381610038161003816 le

10038161003816100381610038161003816100381610038161003816

1198871199011199021

119887 (1 minus 119887) + 119887119901 minus 119901 + 119899

10038161003816100381610038161003816100381610038161003816

119899minus1

sum

119896=0

10038161003816100381610038161198861198961003816100381610038161003816 (48)

This concludes the proof

Competing Interests

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

Acknowledgments

The work here is supported by AP-2013-009

References

[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000

[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004

[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015

[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983

The Scientific World Journal 7

[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945

[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008

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 7: MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World Journal andcomparingthecoe cientsof 2 gives 2 s 2 ti1+T 1 j= 2 i1+T 1 jsV V (1

The Scientific World Journal 7

[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945

[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008

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 8: MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World Journal andcomparingthecoe cientsof 2 gives 2 s 2 ti1+T 1 j= 2 i1+T 1 jsV V (1

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