Research ArticleFekete-Szegouml Inequalities for Certain Classes ofBiunivalent Functions

Fahsene AltJnkaya and Sibel YalccedilJn

Department of Mathematics Faculty of Arts and Science Uludag University Bursa Turkey

Correspondence should be addressed to Sahsene Altınkaya sahsenealtinkayagmailcom

Received 23 June 2014 Accepted 3 September 2014 Published 9 November 2014

Academic Editor Cedric Join

Copyright copy 2014 S Altınkaya and S Yalcın 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 obtain the Fekete-Szego inequalities for the classes 119878lowast119878Σ(120572 120601) and m

119878Σ(120572 120601) of biunivalent functions denoted by subordination

The results presented in this paper improve the recent work of Crisan (2013)

1 Introduction and Definitions

Let 119860 denote the class of analytic functions in the unit disk

119880 = 119911 isin C |119911| lt 1 (1)

that have the form

119891 (119911) = 119911+

infin

sum

119899=2

119886119899119911119899

(2)

Further by 119878 we will denote the class of all functions in 119860which are univalent in 119880

The Koebe one-quarter theorem [1] states that the imageof 119880 under every function 119891 from 119878 contains a disk of radius14 Thus every such univalent function has an inverse 119891minus1which satisfies

119891minus1

(119891 (119911)) = 119911 (119911 isin 119880)

119891 (119891minus1

(119908)) = 119908 (|119908| lt 1199030(119891) 119903

0(119891) ge

1

4)

(3)

where

119891minus1

(119908) = 119908 minus 11988621199082

+ (21198862

2minus 1198863)1199083

minus (51198863

2minus 511988621198863+ 1198864)1199084

+ sdot sdot sdot

(4)

A function 119891 (119911) isin 119860 is said to be biunivalent in 119880 ifboth 119891 (119911) and 119891

minus1

(119911) are univalent in119880 Let Σ denote theclass of biunivalent functions defined in the unit disk 119880

If the functions 119891 and 119892 are analytic in 119880 then 119891 is saidto be subordinate to 119892 written as

119891 (119911) ≺ 119892 (119911) (119911 isin 119880) (5)

if there exists a Schwarz function 119908 (119911) analytic in 119880 with

119908 (0) = 0 |119908 (119911)| lt 1 (119911 isin 119880) (6)

such that

119891 (119911) = 119892 (119908 (119911)) (119911 isin 119880) (7)

Lewin [2] studied the class of biunivalent functionsobtaining the bound 151 for modulus of the second coeffi-cient 10038161003816100381610038161198862

1003816100381610038161003816 Subsequently Brannan and Clunie [3] conjecturedthat 10038161003816100381610038161198862

1003816100381610038161003816 le radic2 for 119891 isin Σ Netanyahu [4] showed thatmax 10038161003816100381610038161198862

1003816100381610038161003816 = 43 if 119891 (119911) isin Σ Brannan and Taha [5] introducedcertain subclasses of the biunivalent function class Σ similarto the familiar subclasses of univalent functions consistingof strongly starlike starlike and convex functions Theyintroduced bistarlike functions and obtained estimates onthe initial coefficients Bounds for the initial coefficients ofseveral classes of functions were also investigated in [6ndash12]The coefficient estimate problem for each of the followingTaylor-Maclaurin coefficients 1003816100381610038161003816119886119899

1003816100381610038161003816 for 119899 isin N 1 2 N =

1 2 3 is presumably still an open problemLet 120601 be an analytic and univalent function with positive

real part in 119880 with 120601 (0) = 1 1206011015840 (0) gt 0 and 120601maps the unitdisk 119880 onto a region starlike with respect to 1 and symmetric

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 327962 6 pageshttpdxdoiorg1011552014327962

2 International Scholarly Research Notices

with respect to the real axis Taylorrsquos series expansion of suchfunction is of the form

120601 (119911) = 1 + 1198611119911 + 11986121199112

+ 11986131199113

+ sdot sdot sdot (8)

where all coefficients are real and 1198611gt 0

By 119878lowast(120601) and 119862(120601) we denote the following classes offunctions

119878lowast

(120601) = 119891 119891 isin 1198601199111198911015840

(119911)

119891 (119911)≺ 120601 (119911) 119911 isin 119880

119862 (120601) = 119891 119891 isin 119860 1 +11991111989110158401015840

(119911)

1198911015840 (119911)≺ 120601 (119911) 119911 isin 119880

(9)

The classes 119878lowast(120601) and 119862(120601) are the extensions of classicalsets of starlike and convex functions and in such a formwere defined and studied by Ma and Minda [13] Theyinvestigated growth and distortion properties of functions in119878lowast

(120601) and 119862(120601) as well as Fekete-Szego inequalities for 119878lowast(120601)and 119862(120601) Their proof of Fekete-Szego inequalities requiresthe univalence of 120601 Ali et al [14] have investigated Fekete-Szego problems for various other classes and their proof doesnot require the univalence or starlikeness of 120601 So in thispaper we assume that 120601 has series expansion 120601(119911) = 1 +

1198611119911 + 119861

21199112

+ sdot sdot sdot 1198611 1198612are real and 119861

1gt 0 A function

119891 is bistarlike of Ma-Minda type or biconvex of Ma-Mindatype if both 119891 and 119891minus1 are respectively Ma-Minda starlike orconvex These classes are denoted respectively by 119878lowast

Σ(120601) and

119862Σ(120601) (see [15])In [16] Sakaguchi introduced the class 119878lowast

119878of starlike

functionswith respect to symmetric points in119880 consisting offunctions 119891 isin 119860 that satisfy the condition Re(1199111198911015840(119911)(119891(119911) minus119891(minus119911))) gt 0 119911 isin 119880 Similarly in [17] Wang et al introducedthe class 119862

119878of convex functions with respect to symmetric

points in 119880 consisting of functions 119891 isin 119860 that satisfy thecondition Re((1199111198911015840(119911))1015840(1198911015840(119911) + 1198911015840(minus119911))) gt 0 119911 isin 119880 In thestyle of Ma and Minda Ravichandran (see [18]) defined theclasses 119878lowast

119878(120601) and 119862

119878(120601)

A function 119891 isin 119860 is in the class 119878lowast119878(120601) if

21199111198911015840

(119911)

119891 (119911) minus 119891 (minus119911)≺ 120601 (119911) 119911 isin 119880 (10)

and in the class 119862119878(120601) if

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911)≺ 120601 (119911) 119911 isin 119880 (11)

In this paper motivated by the earlier work of Zaprawa[19] we obtain the Fekete-Szego inequalities for the classes119878lowast

119878Σ(120572 120601) and m

119878Σ(120572 120601) These inequalities will result in

bounds of the third coefficientwhich are in some cases betterthan these obtained in [7]

In order to derive our main results we require thefollowing lemma

Lemma 1 (see [20]) If 119901 (119911) = 1 + 1199011119911 + 11990121199112

+ 11990131199113

+ sdot sdot sdot isan analytic function in 119880 with positive real part then

10038161003816100381610038161199011198991003816100381610038161003816 le 2 (119899 isin N = 1 2 )

100381610038161003816100381610038161003816100381610038161003816

1199012minus1199012

1

2

100381610038161003816100381610038161003816100381610038161003816

le 2 minus

100381610038161003816100381611990121003816100381610038161003816

2

2

(12)

2 Fekete-Szegouml Inequalities forthe Function Class 119878lowast

119878Σ(120572120601)

Definition 2 (see [7]) A function 119891 isin Σ is said to be in theclass 119878lowast

119878Σ(120572 120601) if the following subordination holds

2 [(1 minus 120572) 1199111198911015840

(119911) + 120572119911 (1199111198911015840

(119911))1015840

]

(1 minus 120572) (119891 (119911) minus 119891 (minus119911)) + 120572119911 (1198911015840 (119911) + 1198911015840 (minus119911))≺ 120601 (119911)

2 [(1 minus 120572)1199081198921015840

(119908) + 120572119908 (1199081198921015840

(119908))1015840

]

(1 minus 120572) (119892 (119908) minus 119892 (minus119908)) + 120572119908 (1198921015840 (119908) + 1198921015840 (minus119908))≺ 120601 (119908)

(13)

where 119892 (119908) = 119891minus1 (119908)

We note that for 120572 = 0 the class 119878lowast119878Σ(120572 120601) reduces to the

class 119878lowast119878(120601) introduced by Ravichandran [18]

Theorem 3 Let 119891 given by (2) be in the class 119878lowast119878Σ(120572 120601) and

120583 isin R Then

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816

le

1198611

2 (1 + 2120572)

1003816100381610038161003816120583 minus 11003816100381610038161003816

le1

1 + 2120572

100381610038161003816100381610038161003816100381610038161003816

1 + 2120572 + 2 (1 + 120572)2(1198611minus 1198612)

1198612

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 minus 120583

1003816100381610038161003816 1198613

1

100381610038161003816100381610038162 (1 + 2120572) 119861

2

1+ 4 (1 + 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1003816100381610038161003816120583 minus 11003816100381610038161003816

ge1

1 + 2120572

100381610038161003816100381610038161003816100381610038161003816

1 + 2120572 + 2 (1 + 120572)2(1198611minus 1198612)

1198612

1

100381610038161003816100381610038161003816100381610038161003816

(14)

Let 119891 isin 119878lowast119878Σ(120572 120601) and 119892 be the analytic extension of 119891minus1

to 119880 Then there exist two functions 119906 and V analytic in 119880

International Scholarly Research Notices 3

with 119906 (0) = V (0) = 0 |119906 (119911)| lt 1 |V (119908)| lt 1 119911 119908 isin 119880 suchthat

2 (1199111198911015840

(119911) + 1205721199112

11989110158401015840

(119911))

(1 minus 120572) (119891 (119911) minus 119891 (minus119911)) + 120572119911 (1198911015840 (119911) + 1198911015840 (minus119911))= 120601 (119906 (119911))

(119911 isin 119880)

2 (1199081198921015840

(119908) + 1205721199082

11989210158401015840

(119908))

(1 minus 120572) (119892 (119908) minus 119892 (minus119908)) + 120572119908 (1198921015840 (119908) + 1198921015840 (minus119908))

= 120601 (V (119908)) (119908 isin 119880)

(15)

Next define the functions 119901 and 119902 by

119901 (119911) =1 + 119906 (119911)

1 minus 119906 (119911)= 1 + 119901

1119911 + 11990121199112

+ sdot sdot sdot

119902 (119908) =1 + V (119908)1 minus V (119908)

= 1 + 1199021119908 + 11990221199082

+ sdot sdot sdot

(16)

Clearly Re119901 (119911) gt 0 and Re 119902 (119908) gt 0 From (16) one canderive

119906 (119911) =119901 (119911) minus 1

119901 (119911) + 1=1

21199011119911 +

1

2(1199012minus1

21199012

1) 1199112

+ sdot sdot sdot

V (119908) =119902 (119908) minus 1

119902 (119908) + 1=1

21199021119908 +

1

2(1199022minus1

21199022

1)1199082

+ sdot sdot sdot

(17)

Combining (8) (15) and (17)

2 (1199111198911015840

(119911) + 1205721199112

11989110158401015840

(119911))

(1 minus 120572) (119891 (119911) minus 119891 (minus119911)) + 120572119911 (1198911015840 (119911) + 1198911015840 (minus119911))

= 1 +1

211986111199011119911 + (

1

411986121199012

1+1

21198611(1199012minus1

21199012

1)) 1199112

+ sdot sdot sdot

2 (1199081198921015840

(119908) + 1205721199082

11989210158401015840

(119908))

(1 minus 120572) (119892 (119908) minus 119892 (minus119908)) + 120572119908 (1198921015840 (119908) + 1198921015840 (minus119908))

= 1 +1

211986111199021119908 + (

1

411986121199022

1+1

21198611(1199022minus1

21199022

1))1199082

+ sdot sdot sdot

(18)

From (18) we deduce

2 (1 + 120572) 1198862=1

211986111199011 (19)

2 (1 + 2120572) 1198863=1

411986121199012

1+1

21198611(1199012minus1

21199012

1) (20)

and

minus2 (1 + 120572) 1198862=1

211986111199021 (21)

2 (1 + 2120572) (21198862

2minus 1198863) =

1

411986121199022

1+1

21198611(1199022minus1

21199022

1) (22)

From (19) and (21) we obtain

1199011= minus1199021 (23)

Subtracting (20) from (22) and applying (23) we have

1198863= 1198862

2+

1

8 (1 + 2120572)1198611(1199012minus 1199022) (24)

By adding (20) to (22) we get

4 (1 + 2120572) 1198862

2=1

21198611(1199012+ 1199022) minus

1

4(1198611minus 1198612) (1199012

1+ 1199022

1)

(25)

Combining this with (19) and (21) leads to

1198862

2=

1198613

1(1199012+ 1199022)

8 [(1 + 2120572) 1198612

1+ 2 (1 + 120572)

2

(1198611minus 1198612)]

(26)

From (24) and (26) it follows that

1198863minus 1205831198862

2= 1198611[(ℎ (120583) +

1

8 (1 + 2120572))1199012

+(ℎ (120583) minus1

8 (1 + 2120572)) 1199022]

(27)

where

ℎ (120583) =1198612

1(1 minus 120583)

8 [(1 + 2120572) 1198612

1+ 2 (1 + 120572)

2

(1198611minus 1198612)]

(28)

Then in view of (8) and (12) we conclude that

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816le

1198611

2 (1 + 2120572) 0 le

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 le

1

8 (1 + 2120572)

41198611

1003816100381610038161003816ℎ (120583)1003816100381610038161003816

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 ge

1

8 (1 + 2120572)

(29)

Taking 120583 = 1 or 120583 = 0 we get the following

Corollary 4 If 119891 isin 119878lowast119878Σ(120572 120601) then

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

1198611

2 (1 + 2120572) (30)

Corollary 5 If 119891 isin 119878lowast119878Σ(120572 120601) then

100381610038161003816100381611988631003816100381610038161003816 le

1198611

2 (1 + 2120572)

1198611minus 1198612

1198612

1

isin (minusinfin minus1 + 2120572

(1 + 120572)2] cup [0infin)

1198613

1

100381610038161003816100381610038162 (1 + 2120572) 119861

2

1+ 4 (1 + 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1198611minus 1198612

1198612

1

isin [minus1 + 2120572

(1 + 120572)2 minus

1 + 2120572

2 (1 + 120572)2)

cup (minus1 + 2120572

2 (1 + 120572)2 0]

(31)

4 International Scholarly Research Notices

Corollary 6 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(32)

then inequalities (30) and (31) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

1 + 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

1 + 2120572 (33)

Corollary 7 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(34)

then inequalities (30) and (31) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

1 + 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

1 + 2120572 (35)

Remark 8 Corollaries 6 and 7 provide an improvement of theestimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

3 Fekete-Szegouml Inequalities forthe Function Class m

119878Σ(120572120601)

Definition 9 (see [7]) A function119891 isin Σ is said to be m119878Σ(120572 120601)

if the following subordination holds

(21199111198911015840

(119911)

119891 (119911) minus 119891 (minus119911))

120572

(

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911))

1minus120572

≺ 120601 (119911)

(21199081198921015840

(119908)

119892 (119908) minus 119892 (minus119908))

120572

(

2 (1199081198921015840

(119908))1015840

1198921015840 (119908) + 1198921015840 (minus119908))

1minus120572

≺ 120601 (119908)

(36)

where 119892 (119908) = 119891minus1 (119908)

We note that for 120572 = 0 the class m119878Σ(120572 120601) reduces to the

class 119862119878(120601) introduced by Ravichandran [18]

Theorem 10 Let 119891 given by (2) be in the class m119878Σ(120572 120601) and

120583 isin R Then

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816

le

1198611

2 (3 minus 2120572)

1003816100381610038161003816120583 minus 11003816100381610038161003816 le

1

3 minus 2120572

times

100381610038161003816100381610038161003816100381610038161003816

3 minus 3120572 + 1205722

+ 2 (2 minus 120572)2(1198611minus 1198612)

1198612

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 minus 120583

1003816100381610038161003816 1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1003816100381610038161003816120583 minus 11003816100381610038161003816 ge

1

3 minus 2120572

times

100381610038161003816100381610038161003816100381610038161003816

3 minus 3120572 + 1205722

+ 2 (2 minus 120572)2(1198611minus 1198612)

1198612

1

100381610038161003816100381610038161003816100381610038161003816

(37)

Let 119891 isin m119878Σ(120572 120601) and 119892 be the analytic extension of 119891minus1

to 119880 Then there exist two functions 119906 and V analytic in 119880with 119906 (0) = V (0) = 0 |119906 (119911)| lt 1 |V (119908)| lt 1 119911 119908 isin 119880 suchthat

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911)= 120601 (119906 (119911))

2 (1199081198921015840

(119908))1015840

1198921015840 (119908) + 1198921015840 (minus119908)= 120601 (V (119908))

(38)

From (38) we deduce

2 (2 minus 120572) 1198862=1

211986111199011 (39)

2 (3 minus 2120572) 1198863minus 2120572 (1 minus 120572) 119886

2

2=1

411986121199012

1+1

21198611(1199012minus1

21199012

1)

(40)

and

minus2 (2 minus 120572) 1198862=1

211986111199021 (41)

2 (3 minus 2120572) (21198862

2minus 1198863) minus 2120572 (1 minus 120572) 119886

2

2

=1

411986121199022

1+1

21198611(1199022minus1

21199022

1)

(42)

From (39) and (41) we obtain

1199011= minus1199021 (43)

Subtracting (40) from (42) and applying (43) we have

1198863= 1198862

2+

1

8 (3 minus 2120572)1198611(1199012minus 1199022) (44)

International Scholarly Research Notices 5

By adding (40) to (42) we get

4 (3 minus 3120572 + 1205722

) 1198862

2=1

21198611(1199012+ 1199022) minus

1

4(1198611minus 1198612) (1199012

1+ 1199022

1)

(45)

Combining this with (39) and (41) leads to

1198862

2=

1198613

1(1199012+ 1199022)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(46)

From (44) and (46) it follows that

1198863minus 1205831198862

2= 1198611[(ℎ (120583) +

1

8 (3 minus 2120572))1199012

+(ℎ (120583) minus1

8 (3 minus 2120572)) 1199022]

(47)

where

ℎ (120583) =1198612

1(1 minus 120583)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(48)

Then in view of (8) and (12) we conclude that

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) 0 le

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 le

1

8 (3 minus 2120572)

41198611

1003816100381610038161003816ℎ (120583)1003816100381610038161003816

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 ge

1

8 (3 minus 2120572)

(49)

Taking 120583 = 1 or 120583 = 0 we get the following

Corollary 11 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) (50)

Corollary 12 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381611988631003816100381610038161003816 le

1198611

2 (3 minus 2120572)

1198611minus 1198612

1198612

1

isin (minusinfin120572 minus 3

2 (2 minus 120572)]

cup [120572 (1 minus 120572)

2 (2 minus 120572)2infin)

1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1198611minus 1198612

1198612

1

isin [120572 minus 3

2 (2 minus 120572)3120572 minus 120572

2

minus 3

2 (2 minus 120572)2)

cup (3120572 minus 120572

2

minus 3

2 (2 minus 120572)2120572 (1 minus 120572)

2 (2 minus 120572)2]

(51)

Corollary 13 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(52)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

3 minus 2120572 (53)

Corollary 14 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(54)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

3 minus 3120572 + 1205722 (55)

Remark 15 Corollaries 13 and 14 provide an improvement ofthe estimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

Conflict of Interests

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

References

[1] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[2] M Lewin ldquoOn a coefficient problem for bi-univalent functionsrdquoProceedings of the American Mathematical Society vol 18 pp63ndash68 1967

[3] D A Brannan and J G Clunie ldquoAspects of comtemporarycomplex analysisrdquo in Proceedings of the NATO Advanced StudyInstute Held at University of Durham July 1ndash20 1979 AcademicPress New York NY USA 1980

[4] E Netanyahu ldquoThe minimal distance of the image boundaryfrom the orijin and the second coefficient of a univalentfunction in |119911| lt 1rdquo Archive for Rational Mechanics andAnalysis vol 32 pp 100ndash112 1969

[5] D A Brannan and T S Taha ldquoOn some classes of bi-univalentfunctionsrdquo Studia Universitatis Babes-Bolyai Mathematica vol31 no 2 pp 70ndash77 1986

[6] S Altinkaya and S Yalcin ldquoInitial coefficient bounds for ageneral class of biunivalent functionsrdquo International Journal ofAnalysis vol 2014 Article ID 867871 4 pages 2014

[7] O Crisan ldquoCoefficient estimates for certain subclasses of bi-univalent functionsrdquo General Mathematics Notes vol 16 no 2pp 93ndash102 2013

[8] B A Frasin and M K Aouf ldquoNew subclasses of bi-univalentfunctionsrdquoAppliedMathematics Letters vol 24 no 9 pp 1569ndash1573 2011

[9] B S Keerthi and B Raja ldquoCoefficient inequality for certainnew subclasses of analytic bi-univalent functionsrdquo TheoreticalMathematics amp Applications vol 3 no 1 pp 1ndash10 2013

[10] N Magesh and J Yamini ldquoCoefficient bounds for certain sub-classes of bi-univalent functionsrdquo International MathematicalForum Journal for Theory and Applications vol 8 no 25-28pp 1337ndash1344 2013

6 International Scholarly Research Notices

[11] H M Srivastava A K Mishra and P Gochhayat ldquoCertainsubclasses of analytic and bi-univalent functionsrdquo AppliedMathematics Letters vol 23 no 10 pp 1188ndash1192 2010

[12] Q H Xu Y C Gui andHM Srivastava ldquoCoefficient estimatesfor a certain subclass of analytic and bi-univalent functionsrdquoApplied Mathematics Letters vol 25 no 6 pp 990ndash994 2012

[13] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis (Tianjin 1992) Conference on Proceed-ings Lecture Notes for Analysis I pp 157ndash169 InternationalPress Cambridge Mass USA 1994

[14] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[15] R M Ali S K Lee V Ravichandran and S SupramanianldquoCoefficient estimates for bi-univalent Ma-Minda starlike andconvex functionsrdquo Applied Mathematics Letters vol 25 no 3pp 344ndash351 2012

[16] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[17] Z G Wang C Y Gao and S M Yuan ldquoOn certain subclassesof close-to-convex and quasi-convex functions with respectto k-symmetric pointsrdquo Journal of Mathematical Analysis andApplications vol 322 no 1 pp 97ndash106 2006

[18] V Ravichandran ldquoStarlike and convex functions with respectto conjugate pointsrdquo Acta Mathematica vol 20 no 1 pp 31ndash372004

[19] P Zaprawa ldquoOn the Fekete-Szego problem for classes ofbi-univalent functionsrdquo Bulletin of the Belgian MathematicalSociety Simon Stevin vol 21 no 1 pp 169ndash178 2014

[20] C Pommerenke Univalent Functions Vandenhoeck ampRuprecht Gottingen Germany 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

2 International Scholarly Research Notices

with respect to the real axis Taylorrsquos series expansion of suchfunction is of the form

120601 (119911) = 1 + 1198611119911 + 11986121199112

+ 11986131199113

+ sdot sdot sdot (8)

where all coefficients are real and 1198611gt 0

By 119878lowast(120601) and 119862(120601) we denote the following classes offunctions

119878lowast

(120601) = 119891 119891 isin 1198601199111198911015840

(119911)

119891 (119911)≺ 120601 (119911) 119911 isin 119880

119862 (120601) = 119891 119891 isin 119860 1 +11991111989110158401015840

(119911)

1198911015840 (119911)≺ 120601 (119911) 119911 isin 119880

(9)

The classes 119878lowast(120601) and 119862(120601) are the extensions of classicalsets of starlike and convex functions and in such a formwere defined and studied by Ma and Minda [13] Theyinvestigated growth and distortion properties of functions in119878lowast

(120601) and 119862(120601) as well as Fekete-Szego inequalities for 119878lowast(120601)and 119862(120601) Their proof of Fekete-Szego inequalities requiresthe univalence of 120601 Ali et al [14] have investigated Fekete-Szego problems for various other classes and their proof doesnot require the univalence or starlikeness of 120601 So in thispaper we assume that 120601 has series expansion 120601(119911) = 1 +

1198611119911 + 119861

21199112

+ sdot sdot sdot 1198611 1198612are real and 119861

1gt 0 A function

119891 is bistarlike of Ma-Minda type or biconvex of Ma-Mindatype if both 119891 and 119891minus1 are respectively Ma-Minda starlike orconvex These classes are denoted respectively by 119878lowast

Σ(120601) and

119862Σ(120601) (see [15])In [16] Sakaguchi introduced the class 119878lowast

119878of starlike

functionswith respect to symmetric points in119880 consisting offunctions 119891 isin 119860 that satisfy the condition Re(1199111198911015840(119911)(119891(119911) minus119891(minus119911))) gt 0 119911 isin 119880 Similarly in [17] Wang et al introducedthe class 119862

119878of convex functions with respect to symmetric

points in 119880 consisting of functions 119891 isin 119860 that satisfy thecondition Re((1199111198911015840(119911))1015840(1198911015840(119911) + 1198911015840(minus119911))) gt 0 119911 isin 119880 In thestyle of Ma and Minda Ravichandran (see [18]) defined theclasses 119878lowast

119878(120601) and 119862

119878(120601)

A function 119891 isin 119860 is in the class 119878lowast119878(120601) if

21199111198911015840

(119911)

119891 (119911) minus 119891 (minus119911)≺ 120601 (119911) 119911 isin 119880 (10)

and in the class 119862119878(120601) if

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911)≺ 120601 (119911) 119911 isin 119880 (11)

In this paper motivated by the earlier work of Zaprawa[19] we obtain the Fekete-Szego inequalities for the classes119878lowast

119878Σ(120572 120601) and m

119878Σ(120572 120601) These inequalities will result in

bounds of the third coefficientwhich are in some cases betterthan these obtained in [7]

In order to derive our main results we require thefollowing lemma

Lemma 1 (see [20]) If 119901 (119911) = 1 + 1199011119911 + 11990121199112

+ 11990131199113

+ sdot sdot sdot isan analytic function in 119880 with positive real part then

10038161003816100381610038161199011198991003816100381610038161003816 le 2 (119899 isin N = 1 2 )

100381610038161003816100381610038161003816100381610038161003816

1199012minus1199012

1

2

100381610038161003816100381610038161003816100381610038161003816

le 2 minus

100381610038161003816100381611990121003816100381610038161003816

2

2

(12)

2 Fekete-Szegouml Inequalities forthe Function Class 119878lowast

119878Σ(120572120601)

Definition 2 (see [7]) A function 119891 isin Σ is said to be in theclass 119878lowast

119878Σ(120572 120601) if the following subordination holds

2 [(1 minus 120572) 1199111198911015840

(119911) + 120572119911 (1199111198911015840

(119911))1015840

]

(1 minus 120572) (119891 (119911) minus 119891 (minus119911)) + 120572119911 (1198911015840 (119911) + 1198911015840 (minus119911))≺ 120601 (119911)

2 [(1 minus 120572)1199081198921015840

(119908) + 120572119908 (1199081198921015840

(119908))1015840

]

(1 minus 120572) (119892 (119908) minus 119892 (minus119908)) + 120572119908 (1198921015840 (119908) + 1198921015840 (minus119908))≺ 120601 (119908)

(13)

where 119892 (119908) = 119891minus1 (119908)

We note that for 120572 = 0 the class 119878lowast119878Σ(120572 120601) reduces to the

class 119878lowast119878(120601) introduced by Ravichandran [18]

Theorem 3 Let 119891 given by (2) be in the class 119878lowast119878Σ(120572 120601) and

120583 isin R Then

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816

le

1198611

2 (1 + 2120572)

1003816100381610038161003816120583 minus 11003816100381610038161003816

le1

1 + 2120572

100381610038161003816100381610038161003816100381610038161003816

1 + 2120572 + 2 (1 + 120572)2(1198611minus 1198612)

1198612

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 minus 120583

1003816100381610038161003816 1198613

1

100381610038161003816100381610038162 (1 + 2120572) 119861

2

1+ 4 (1 + 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1003816100381610038161003816120583 minus 11003816100381610038161003816

ge1

1 + 2120572

100381610038161003816100381610038161003816100381610038161003816

1 + 2120572 + 2 (1 + 120572)2(1198611minus 1198612)

1198612

1

100381610038161003816100381610038161003816100381610038161003816

(14)

Let 119891 isin 119878lowast119878Σ(120572 120601) and 119892 be the analytic extension of 119891minus1

to 119880 Then there exist two functions 119906 and V analytic in 119880

International Scholarly Research Notices 3

with 119906 (0) = V (0) = 0 |119906 (119911)| lt 1 |V (119908)| lt 1 119911 119908 isin 119880 suchthat

2 (1199111198911015840

(119911) + 1205721199112

11989110158401015840

(119911))

(1 minus 120572) (119891 (119911) minus 119891 (minus119911)) + 120572119911 (1198911015840 (119911) + 1198911015840 (minus119911))= 120601 (119906 (119911))

(119911 isin 119880)

2 (1199081198921015840

(119908) + 1205721199082

11989210158401015840

(119908))

(1 minus 120572) (119892 (119908) minus 119892 (minus119908)) + 120572119908 (1198921015840 (119908) + 1198921015840 (minus119908))

= 120601 (V (119908)) (119908 isin 119880)

(15)

Next define the functions 119901 and 119902 by

119901 (119911) =1 + 119906 (119911)

1 minus 119906 (119911)= 1 + 119901

1119911 + 11990121199112

+ sdot sdot sdot

119902 (119908) =1 + V (119908)1 minus V (119908)

= 1 + 1199021119908 + 11990221199082

+ sdot sdot sdot

(16)

Clearly Re119901 (119911) gt 0 and Re 119902 (119908) gt 0 From (16) one canderive

119906 (119911) =119901 (119911) minus 1

119901 (119911) + 1=1

21199011119911 +

1

2(1199012minus1

21199012

1) 1199112

+ sdot sdot sdot

V (119908) =119902 (119908) minus 1

119902 (119908) + 1=1

21199021119908 +

1

2(1199022minus1

21199022

1)1199082

+ sdot sdot sdot

(17)

Combining (8) (15) and (17)

2 (1199111198911015840

(119911) + 1205721199112

11989110158401015840

(119911))

(1 minus 120572) (119891 (119911) minus 119891 (minus119911)) + 120572119911 (1198911015840 (119911) + 1198911015840 (minus119911))

= 1 +1

211986111199011119911 + (

1

411986121199012

1+1

21198611(1199012minus1

21199012

1)) 1199112

+ sdot sdot sdot

2 (1199081198921015840

(119908) + 1205721199082

11989210158401015840

(119908))

(1 minus 120572) (119892 (119908) minus 119892 (minus119908)) + 120572119908 (1198921015840 (119908) + 1198921015840 (minus119908))

= 1 +1

211986111199021119908 + (

1

411986121199022

1+1

21198611(1199022minus1

21199022

1))1199082

+ sdot sdot sdot

(18)

From (18) we deduce

2 (1 + 120572) 1198862=1

211986111199011 (19)

2 (1 + 2120572) 1198863=1

411986121199012

1+1

21198611(1199012minus1

21199012

1) (20)

and

minus2 (1 + 120572) 1198862=1

211986111199021 (21)

2 (1 + 2120572) (21198862

2minus 1198863) =

1

411986121199022

1+1

21198611(1199022minus1

21199022

1) (22)

From (19) and (21) we obtain

1199011= minus1199021 (23)

Subtracting (20) from (22) and applying (23) we have

1198863= 1198862

2+

1

8 (1 + 2120572)1198611(1199012minus 1199022) (24)

By adding (20) to (22) we get

4 (1 + 2120572) 1198862

2=1

21198611(1199012+ 1199022) minus

1

4(1198611minus 1198612) (1199012

1+ 1199022

1)

(25)

Combining this with (19) and (21) leads to

1198862

2=

1198613

1(1199012+ 1199022)

8 [(1 + 2120572) 1198612

1+ 2 (1 + 120572)

2

(1198611minus 1198612)]

(26)

From (24) and (26) it follows that

1198863minus 1205831198862

2= 1198611[(ℎ (120583) +

1

8 (1 + 2120572))1199012

+(ℎ (120583) minus1

8 (1 + 2120572)) 1199022]

(27)

where

ℎ (120583) =1198612

1(1 minus 120583)

8 [(1 + 2120572) 1198612

1+ 2 (1 + 120572)

2

(1198611minus 1198612)]

(28)

Then in view of (8) and (12) we conclude that

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816le

1198611

2 (1 + 2120572) 0 le

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 le

1

8 (1 + 2120572)

41198611

1003816100381610038161003816ℎ (120583)1003816100381610038161003816

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 ge

1

8 (1 + 2120572)

(29)

Taking 120583 = 1 or 120583 = 0 we get the following

Corollary 4 If 119891 isin 119878lowast119878Σ(120572 120601) then

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

1198611

2 (1 + 2120572) (30)

Corollary 5 If 119891 isin 119878lowast119878Σ(120572 120601) then

100381610038161003816100381611988631003816100381610038161003816 le

1198611

2 (1 + 2120572)

1198611minus 1198612

1198612

1

isin (minusinfin minus1 + 2120572

(1 + 120572)2] cup [0infin)

1198613

1

100381610038161003816100381610038162 (1 + 2120572) 119861

2

1+ 4 (1 + 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1198611minus 1198612

1198612

1

isin [minus1 + 2120572

(1 + 120572)2 minus

1 + 2120572

2 (1 + 120572)2)

cup (minus1 + 2120572

2 (1 + 120572)2 0]

(31)

4 International Scholarly Research Notices

Corollary 6 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(32)

then inequalities (30) and (31) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

1 + 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

1 + 2120572 (33)

Corollary 7 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(34)

then inequalities (30) and (31) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

1 + 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

1 + 2120572 (35)

Remark 8 Corollaries 6 and 7 provide an improvement of theestimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

3 Fekete-Szegouml Inequalities forthe Function Class m

119878Σ(120572120601)

Definition 9 (see [7]) A function119891 isin Σ is said to be m119878Σ(120572 120601)

if the following subordination holds

(21199111198911015840

(119911)

119891 (119911) minus 119891 (minus119911))

120572

(

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911))

1minus120572

≺ 120601 (119911)

(21199081198921015840

(119908)

119892 (119908) minus 119892 (minus119908))

120572

(

2 (1199081198921015840

(119908))1015840

1198921015840 (119908) + 1198921015840 (minus119908))

1minus120572

≺ 120601 (119908)

(36)

where 119892 (119908) = 119891minus1 (119908)

We note that for 120572 = 0 the class m119878Σ(120572 120601) reduces to the

class 119862119878(120601) introduced by Ravichandran [18]

Theorem 10 Let 119891 given by (2) be in the class m119878Σ(120572 120601) and

120583 isin R Then

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816

le

1198611

2 (3 minus 2120572)

1003816100381610038161003816120583 minus 11003816100381610038161003816 le

1

3 minus 2120572

times

100381610038161003816100381610038161003816100381610038161003816

3 minus 3120572 + 1205722

+ 2 (2 minus 120572)2(1198611minus 1198612)

1198612

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 minus 120583

1003816100381610038161003816 1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1003816100381610038161003816120583 minus 11003816100381610038161003816 ge

1

3 minus 2120572

times

100381610038161003816100381610038161003816100381610038161003816

3 minus 3120572 + 1205722

+ 2 (2 minus 120572)2(1198611minus 1198612)

1198612

1

100381610038161003816100381610038161003816100381610038161003816

(37)

Let 119891 isin m119878Σ(120572 120601) and 119892 be the analytic extension of 119891minus1

to 119880 Then there exist two functions 119906 and V analytic in 119880with 119906 (0) = V (0) = 0 |119906 (119911)| lt 1 |V (119908)| lt 1 119911 119908 isin 119880 suchthat

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911)= 120601 (119906 (119911))

2 (1199081198921015840

(119908))1015840

1198921015840 (119908) + 1198921015840 (minus119908)= 120601 (V (119908))

(38)

From (38) we deduce

2 (2 minus 120572) 1198862=1

211986111199011 (39)

2 (3 minus 2120572) 1198863minus 2120572 (1 minus 120572) 119886

2

2=1

411986121199012

1+1

21198611(1199012minus1

21199012

1)

(40)

and

minus2 (2 minus 120572) 1198862=1

211986111199021 (41)

2 (3 minus 2120572) (21198862

2minus 1198863) minus 2120572 (1 minus 120572) 119886

2

2

=1

411986121199022

1+1

21198611(1199022minus1

21199022

1)

(42)

From (39) and (41) we obtain

1199011= minus1199021 (43)

Subtracting (40) from (42) and applying (43) we have

1198863= 1198862

2+

1

8 (3 minus 2120572)1198611(1199012minus 1199022) (44)

International Scholarly Research Notices 5

By adding (40) to (42) we get

4 (3 minus 3120572 + 1205722

) 1198862

2=1

21198611(1199012+ 1199022) minus

1

4(1198611minus 1198612) (1199012

1+ 1199022

1)

(45)

Combining this with (39) and (41) leads to

1198862

2=

1198613

1(1199012+ 1199022)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(46)

From (44) and (46) it follows that

1198863minus 1205831198862

2= 1198611[(ℎ (120583) +

1

8 (3 minus 2120572))1199012

+(ℎ (120583) minus1

8 (3 minus 2120572)) 1199022]

(47)

where

ℎ (120583) =1198612

1(1 minus 120583)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(48)

Then in view of (8) and (12) we conclude that

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) 0 le

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 le

1

8 (3 minus 2120572)

41198611

1003816100381610038161003816ℎ (120583)1003816100381610038161003816

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 ge

1

8 (3 minus 2120572)

(49)

Taking 120583 = 1 or 120583 = 0 we get the following

Corollary 11 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) (50)

Corollary 12 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381611988631003816100381610038161003816 le

1198611

2 (3 minus 2120572)

1198611minus 1198612

1198612

1

isin (minusinfin120572 minus 3

2 (2 minus 120572)]

cup [120572 (1 minus 120572)

2 (2 minus 120572)2infin)

1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1198611minus 1198612

1198612

1

isin [120572 minus 3

2 (2 minus 120572)3120572 minus 120572

2

minus 3

2 (2 minus 120572)2)

cup (3120572 minus 120572

2

minus 3

2 (2 minus 120572)2120572 (1 minus 120572)

2 (2 minus 120572)2]

(51)

Corollary 13 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(52)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

3 minus 2120572 (53)

Corollary 14 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(54)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

3 minus 3120572 + 1205722 (55)

Remark 15 Corollaries 13 and 14 provide an improvement ofthe estimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

Conflict of Interests

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

References

[1] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[2] M Lewin ldquoOn a coefficient problem for bi-univalent functionsrdquoProceedings of the American Mathematical Society vol 18 pp63ndash68 1967

[3] D A Brannan and J G Clunie ldquoAspects of comtemporarycomplex analysisrdquo in Proceedings of the NATO Advanced StudyInstute Held at University of Durham July 1ndash20 1979 AcademicPress New York NY USA 1980

[4] E Netanyahu ldquoThe minimal distance of the image boundaryfrom the orijin and the second coefficient of a univalentfunction in |119911| lt 1rdquo Archive for Rational Mechanics andAnalysis vol 32 pp 100ndash112 1969

[5] D A Brannan and T S Taha ldquoOn some classes of bi-univalentfunctionsrdquo Studia Universitatis Babes-Bolyai Mathematica vol31 no 2 pp 70ndash77 1986

[6] S Altinkaya and S Yalcin ldquoInitial coefficient bounds for ageneral class of biunivalent functionsrdquo International Journal ofAnalysis vol 2014 Article ID 867871 4 pages 2014

[7] O Crisan ldquoCoefficient estimates for certain subclasses of bi-univalent functionsrdquo General Mathematics Notes vol 16 no 2pp 93ndash102 2013

[8] B A Frasin and M K Aouf ldquoNew subclasses of bi-univalentfunctionsrdquoAppliedMathematics Letters vol 24 no 9 pp 1569ndash1573 2011

[9] B S Keerthi and B Raja ldquoCoefficient inequality for certainnew subclasses of analytic bi-univalent functionsrdquo TheoreticalMathematics amp Applications vol 3 no 1 pp 1ndash10 2013

[10] N Magesh and J Yamini ldquoCoefficient bounds for certain sub-classes of bi-univalent functionsrdquo International MathematicalForum Journal for Theory and Applications vol 8 no 25-28pp 1337ndash1344 2013

6 International Scholarly Research Notices

[11] H M Srivastava A K Mishra and P Gochhayat ldquoCertainsubclasses of analytic and bi-univalent functionsrdquo AppliedMathematics Letters vol 23 no 10 pp 1188ndash1192 2010

[12] Q H Xu Y C Gui andHM Srivastava ldquoCoefficient estimatesfor a certain subclass of analytic and bi-univalent functionsrdquoApplied Mathematics Letters vol 25 no 6 pp 990ndash994 2012

[13] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis (Tianjin 1992) Conference on Proceed-ings Lecture Notes for Analysis I pp 157ndash169 InternationalPress Cambridge Mass USA 1994

[14] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[15] R M Ali S K Lee V Ravichandran and S SupramanianldquoCoefficient estimates for bi-univalent Ma-Minda starlike andconvex functionsrdquo Applied Mathematics Letters vol 25 no 3pp 344ndash351 2012

[16] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[17] Z G Wang C Y Gao and S M Yuan ldquoOn certain subclassesof close-to-convex and quasi-convex functions with respectto k-symmetric pointsrdquo Journal of Mathematical Analysis andApplications vol 322 no 1 pp 97ndash106 2006

[18] V Ravichandran ldquoStarlike and convex functions with respectto conjugate pointsrdquo Acta Mathematica vol 20 no 1 pp 31ndash372004

[19] P Zaprawa ldquoOn the Fekete-Szego problem for classes ofbi-univalent functionsrdquo Bulletin of the Belgian MathematicalSociety Simon Stevin vol 21 no 1 pp 169ndash178 2014

[20] C Pommerenke Univalent Functions Vandenhoeck ampRuprecht Gottingen Germany 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 3

with 119906 (0) = V (0) = 0 |119906 (119911)| lt 1 |V (119908)| lt 1 119911 119908 isin 119880 suchthat

2 (1199111198911015840

(119911) + 1205721199112

11989110158401015840

(119911))

(1 minus 120572) (119891 (119911) minus 119891 (minus119911)) + 120572119911 (1198911015840 (119911) + 1198911015840 (minus119911))= 120601 (119906 (119911))

(119911 isin 119880)

2 (1199081198921015840

(119908) + 1205721199082

11989210158401015840

(119908))

(1 minus 120572) (119892 (119908) minus 119892 (minus119908)) + 120572119908 (1198921015840 (119908) + 1198921015840 (minus119908))

= 120601 (V (119908)) (119908 isin 119880)

(15)

Next define the functions 119901 and 119902 by

119901 (119911) =1 + 119906 (119911)

1 minus 119906 (119911)= 1 + 119901

1119911 + 11990121199112

+ sdot sdot sdot

119902 (119908) =1 + V (119908)1 minus V (119908)

= 1 + 1199021119908 + 11990221199082

+ sdot sdot sdot

(16)

Clearly Re119901 (119911) gt 0 and Re 119902 (119908) gt 0 From (16) one canderive

119906 (119911) =119901 (119911) minus 1

119901 (119911) + 1=1

21199011119911 +

1

2(1199012minus1

21199012

1) 1199112

+ sdot sdot sdot

V (119908) =119902 (119908) minus 1

119902 (119908) + 1=1

21199021119908 +

1

2(1199022minus1

21199022

1)1199082

+ sdot sdot sdot

(17)

Combining (8) (15) and (17)

2 (1199111198911015840

(119911) + 1205721199112

11989110158401015840

(119911))

(1 minus 120572) (119891 (119911) minus 119891 (minus119911)) + 120572119911 (1198911015840 (119911) + 1198911015840 (minus119911))

= 1 +1

211986111199011119911 + (

1

411986121199012

1+1

21198611(1199012minus1

21199012

1)) 1199112

+ sdot sdot sdot

2 (1199081198921015840

(119908) + 1205721199082

11989210158401015840

(119908))

(1 minus 120572) (119892 (119908) minus 119892 (minus119908)) + 120572119908 (1198921015840 (119908) + 1198921015840 (minus119908))

= 1 +1

211986111199021119908 + (

1

411986121199022

1+1

21198611(1199022minus1

21199022

1))1199082

+ sdot sdot sdot

(18)

From (18) we deduce

2 (1 + 120572) 1198862=1

211986111199011 (19)

2 (1 + 2120572) 1198863=1

411986121199012

1+1

21198611(1199012minus1

21199012

1) (20)

and

minus2 (1 + 120572) 1198862=1

211986111199021 (21)

2 (1 + 2120572) (21198862

2minus 1198863) =

1

411986121199022

1+1

21198611(1199022minus1

21199022

1) (22)

From (19) and (21) we obtain

1199011= minus1199021 (23)

Subtracting (20) from (22) and applying (23) we have

1198863= 1198862

2+

1

8 (1 + 2120572)1198611(1199012minus 1199022) (24)

By adding (20) to (22) we get

4 (1 + 2120572) 1198862

2=1

21198611(1199012+ 1199022) minus

1

4(1198611minus 1198612) (1199012

1+ 1199022

1)

(25)

Combining this with (19) and (21) leads to

1198862

2=

1198613

1(1199012+ 1199022)

8 [(1 + 2120572) 1198612

1+ 2 (1 + 120572)

2

(1198611minus 1198612)]

(26)

From (24) and (26) it follows that

1198863minus 1205831198862

2= 1198611[(ℎ (120583) +

1

8 (1 + 2120572))1199012

+(ℎ (120583) minus1

8 (1 + 2120572)) 1199022]

(27)

where

ℎ (120583) =1198612

1(1 minus 120583)

8 [(1 + 2120572) 1198612

1+ 2 (1 + 120572)

2

(1198611minus 1198612)]

(28)

Then in view of (8) and (12) we conclude that

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816le

1198611

2 (1 + 2120572) 0 le

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 le

1

8 (1 + 2120572)

41198611

1003816100381610038161003816ℎ (120583)1003816100381610038161003816

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 ge

1

8 (1 + 2120572)

(29)

Taking 120583 = 1 or 120583 = 0 we get the following

Corollary 4 If 119891 isin 119878lowast119878Σ(120572 120601) then

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

1198611

2 (1 + 2120572) (30)

Corollary 5 If 119891 isin 119878lowast119878Σ(120572 120601) then

100381610038161003816100381611988631003816100381610038161003816 le

1198611

2 (1 + 2120572)

1198611minus 1198612

1198612

1

isin (minusinfin minus1 + 2120572

(1 + 120572)2] cup [0infin)

1198613

1

100381610038161003816100381610038162 (1 + 2120572) 119861

2

1+ 4 (1 + 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1198611minus 1198612

1198612

1

isin [minus1 + 2120572

(1 + 120572)2 minus

1 + 2120572

2 (1 + 120572)2)

cup (minus1 + 2120572

2 (1 + 120572)2 0]

(31)

4 International Scholarly Research Notices

Corollary 6 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(32)

then inequalities (30) and (31) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

1 + 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

1 + 2120572 (33)

Corollary 7 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(34)

then inequalities (30) and (31) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

1 + 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

1 + 2120572 (35)

Remark 8 Corollaries 6 and 7 provide an improvement of theestimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

3 Fekete-Szegouml Inequalities forthe Function Class m

119878Σ(120572120601)

Definition 9 (see [7]) A function119891 isin Σ is said to be m119878Σ(120572 120601)

if the following subordination holds

(21199111198911015840

(119911)

119891 (119911) minus 119891 (minus119911))

120572

(

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911))

1minus120572

≺ 120601 (119911)

(21199081198921015840

(119908)

119892 (119908) minus 119892 (minus119908))

120572

(

2 (1199081198921015840

(119908))1015840

1198921015840 (119908) + 1198921015840 (minus119908))

1minus120572

≺ 120601 (119908)

(36)

where 119892 (119908) = 119891minus1 (119908)

We note that for 120572 = 0 the class m119878Σ(120572 120601) reduces to the

class 119862119878(120601) introduced by Ravichandran [18]

Theorem 10 Let 119891 given by (2) be in the class m119878Σ(120572 120601) and

120583 isin R Then

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816

le

1198611

2 (3 minus 2120572)

1003816100381610038161003816120583 minus 11003816100381610038161003816 le

1

3 minus 2120572

times

100381610038161003816100381610038161003816100381610038161003816

3 minus 3120572 + 1205722

+ 2 (2 minus 120572)2(1198611minus 1198612)

1198612

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 minus 120583

1003816100381610038161003816 1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1003816100381610038161003816120583 minus 11003816100381610038161003816 ge

1

3 minus 2120572

times

100381610038161003816100381610038161003816100381610038161003816

3 minus 3120572 + 1205722

+ 2 (2 minus 120572)2(1198611minus 1198612)

1198612

1

100381610038161003816100381610038161003816100381610038161003816

(37)

Let 119891 isin m119878Σ(120572 120601) and 119892 be the analytic extension of 119891minus1

to 119880 Then there exist two functions 119906 and V analytic in 119880with 119906 (0) = V (0) = 0 |119906 (119911)| lt 1 |V (119908)| lt 1 119911 119908 isin 119880 suchthat

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911)= 120601 (119906 (119911))

2 (1199081198921015840

(119908))1015840

1198921015840 (119908) + 1198921015840 (minus119908)= 120601 (V (119908))

(38)

From (38) we deduce

2 (2 minus 120572) 1198862=1

211986111199011 (39)

2 (3 minus 2120572) 1198863minus 2120572 (1 minus 120572) 119886

2

2=1

411986121199012

1+1

21198611(1199012minus1

21199012

1)

(40)

and

minus2 (2 minus 120572) 1198862=1

211986111199021 (41)

2 (3 minus 2120572) (21198862

2minus 1198863) minus 2120572 (1 minus 120572) 119886

2

2

=1

411986121199022

1+1

21198611(1199022minus1

21199022

1)

(42)

From (39) and (41) we obtain

1199011= minus1199021 (43)

Subtracting (40) from (42) and applying (43) we have

1198863= 1198862

2+

1

8 (3 minus 2120572)1198611(1199012minus 1199022) (44)

International Scholarly Research Notices 5

By adding (40) to (42) we get

4 (3 minus 3120572 + 1205722

) 1198862

2=1

21198611(1199012+ 1199022) minus

1

4(1198611minus 1198612) (1199012

1+ 1199022

1)

(45)

Combining this with (39) and (41) leads to

1198862

2=

1198613

1(1199012+ 1199022)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(46)

From (44) and (46) it follows that

1198863minus 1205831198862

2= 1198611[(ℎ (120583) +

1

8 (3 minus 2120572))1199012

+(ℎ (120583) minus1

8 (3 minus 2120572)) 1199022]

(47)

where

ℎ (120583) =1198612

1(1 minus 120583)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(48)

Then in view of (8) and (12) we conclude that

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) 0 le

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 le

1

8 (3 minus 2120572)

41198611

1003816100381610038161003816ℎ (120583)1003816100381610038161003816

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 ge

1

8 (3 minus 2120572)

(49)

Taking 120583 = 1 or 120583 = 0 we get the following

Corollary 11 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) (50)

Corollary 12 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381611988631003816100381610038161003816 le

1198611

2 (3 minus 2120572)

1198611minus 1198612

1198612

1

isin (minusinfin120572 minus 3

2 (2 minus 120572)]

cup [120572 (1 minus 120572)

2 (2 minus 120572)2infin)

1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1198611minus 1198612

1198612

1

isin [120572 minus 3

2 (2 minus 120572)3120572 minus 120572

2

minus 3

2 (2 minus 120572)2)

cup (3120572 minus 120572

2

minus 3

2 (2 minus 120572)2120572 (1 minus 120572)

2 (2 minus 120572)2]

(51)

Corollary 13 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(52)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

3 minus 2120572 (53)

Corollary 14 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(54)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

3 minus 3120572 + 1205722 (55)

Remark 15 Corollaries 13 and 14 provide an improvement ofthe estimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

Conflict of Interests

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

References

[1] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[2] M Lewin ldquoOn a coefficient problem for bi-univalent functionsrdquoProceedings of the American Mathematical Society vol 18 pp63ndash68 1967

[3] D A Brannan and J G Clunie ldquoAspects of comtemporarycomplex analysisrdquo in Proceedings of the NATO Advanced StudyInstute Held at University of Durham July 1ndash20 1979 AcademicPress New York NY USA 1980

[4] E Netanyahu ldquoThe minimal distance of the image boundaryfrom the orijin and the second coefficient of a univalentfunction in |119911| lt 1rdquo Archive for Rational Mechanics andAnalysis vol 32 pp 100ndash112 1969

[5] D A Brannan and T S Taha ldquoOn some classes of bi-univalentfunctionsrdquo Studia Universitatis Babes-Bolyai Mathematica vol31 no 2 pp 70ndash77 1986

[6] S Altinkaya and S Yalcin ldquoInitial coefficient bounds for ageneral class of biunivalent functionsrdquo International Journal ofAnalysis vol 2014 Article ID 867871 4 pages 2014

[7] O Crisan ldquoCoefficient estimates for certain subclasses of bi-univalent functionsrdquo General Mathematics Notes vol 16 no 2pp 93ndash102 2013

[8] B A Frasin and M K Aouf ldquoNew subclasses of bi-univalentfunctionsrdquoAppliedMathematics Letters vol 24 no 9 pp 1569ndash1573 2011

[9] B S Keerthi and B Raja ldquoCoefficient inequality for certainnew subclasses of analytic bi-univalent functionsrdquo TheoreticalMathematics amp Applications vol 3 no 1 pp 1ndash10 2013

[10] N Magesh and J Yamini ldquoCoefficient bounds for certain sub-classes of bi-univalent functionsrdquo International MathematicalForum Journal for Theory and Applications vol 8 no 25-28pp 1337ndash1344 2013

6 International Scholarly Research Notices

[11] H M Srivastava A K Mishra and P Gochhayat ldquoCertainsubclasses of analytic and bi-univalent functionsrdquo AppliedMathematics Letters vol 23 no 10 pp 1188ndash1192 2010

[12] Q H Xu Y C Gui andHM Srivastava ldquoCoefficient estimatesfor a certain subclass of analytic and bi-univalent functionsrdquoApplied Mathematics Letters vol 25 no 6 pp 990ndash994 2012

[13] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis (Tianjin 1992) Conference on Proceed-ings Lecture Notes for Analysis I pp 157ndash169 InternationalPress Cambridge Mass USA 1994

[14] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[15] R M Ali S K Lee V Ravichandran and S SupramanianldquoCoefficient estimates for bi-univalent Ma-Minda starlike andconvex functionsrdquo Applied Mathematics Letters vol 25 no 3pp 344ndash351 2012

[16] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[17] Z G Wang C Y Gao and S M Yuan ldquoOn certain subclassesof close-to-convex and quasi-convex functions with respectto k-symmetric pointsrdquo Journal of Mathematical Analysis andApplications vol 322 no 1 pp 97ndash106 2006

[18] V Ravichandran ldquoStarlike and convex functions with respectto conjugate pointsrdquo Acta Mathematica vol 20 no 1 pp 31ndash372004

[19] P Zaprawa ldquoOn the Fekete-Szego problem for classes ofbi-univalent functionsrdquo Bulletin of the Belgian MathematicalSociety Simon Stevin vol 21 no 1 pp 169ndash178 2014

[20] C Pommerenke Univalent Functions Vandenhoeck ampRuprecht Gottingen Germany 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Scholarly Research Notices

Corollary 6 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(32)

then inequalities (30) and (31) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

1 + 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

1 + 2120572 (33)

Corollary 7 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(34)

then inequalities (30) and (31) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

1 + 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

1 + 2120572 (35)

Remark 8 Corollaries 6 and 7 provide an improvement of theestimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

3 Fekete-Szegouml Inequalities forthe Function Class m

119878Σ(120572120601)

Definition 9 (see [7]) A function119891 isin Σ is said to be m119878Σ(120572 120601)

if the following subordination holds

(21199111198911015840

(119911)

119891 (119911) minus 119891 (minus119911))

120572

(

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911))

1minus120572

≺ 120601 (119911)

(21199081198921015840

(119908)

119892 (119908) minus 119892 (minus119908))

120572

(

2 (1199081198921015840

(119908))1015840

1198921015840 (119908) + 1198921015840 (minus119908))

1minus120572

≺ 120601 (119908)

(36)

where 119892 (119908) = 119891minus1 (119908)

We note that for 120572 = 0 the class m119878Σ(120572 120601) reduces to the

class 119862119878(120601) introduced by Ravichandran [18]

Theorem 10 Let 119891 given by (2) be in the class m119878Σ(120572 120601) and

120583 isin R Then

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816

le

1198611

2 (3 minus 2120572)

1003816100381610038161003816120583 minus 11003816100381610038161003816 le

1

3 minus 2120572

times

100381610038161003816100381610038161003816100381610038161003816

3 minus 3120572 + 1205722

+ 2 (2 minus 120572)2(1198611minus 1198612)

1198612

1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161 minus 120583

1003816100381610038161003816 1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1003816100381610038161003816120583 minus 11003816100381610038161003816 ge

1

3 minus 2120572

times

100381610038161003816100381610038161003816100381610038161003816

3 minus 3120572 + 1205722

+ 2 (2 minus 120572)2(1198611minus 1198612)

1198612

1

100381610038161003816100381610038161003816100381610038161003816

(37)

Let 119891 isin m119878Σ(120572 120601) and 119892 be the analytic extension of 119891minus1

to 119880 Then there exist two functions 119906 and V analytic in 119880with 119906 (0) = V (0) = 0 |119906 (119911)| lt 1 |V (119908)| lt 1 119911 119908 isin 119880 suchthat

2 (1199111198911015840

(119911))1015840

1198911015840 (119911) + 1198911015840 (minus119911)= 120601 (119906 (119911))

2 (1199081198921015840

(119908))1015840

1198921015840 (119908) + 1198921015840 (minus119908)= 120601 (V (119908))

(38)

From (38) we deduce

2 (2 minus 120572) 1198862=1

211986111199011 (39)

2 (3 minus 2120572) 1198863minus 2120572 (1 minus 120572) 119886

2

2=1

411986121199012

1+1

21198611(1199012minus1

21199012

1)

(40)

and

minus2 (2 minus 120572) 1198862=1

211986111199021 (41)

2 (3 minus 2120572) (21198862

2minus 1198863) minus 2120572 (1 minus 120572) 119886

2

2

=1

411986121199022

1+1

21198611(1199022minus1

21199022

1)

(42)

From (39) and (41) we obtain

1199011= minus1199021 (43)

Subtracting (40) from (42) and applying (43) we have

1198863= 1198862

2+

1

8 (3 minus 2120572)1198611(1199012minus 1199022) (44)

International Scholarly Research Notices 5

By adding (40) to (42) we get

4 (3 minus 3120572 + 1205722

) 1198862

2=1

21198611(1199012+ 1199022) minus

1

4(1198611minus 1198612) (1199012

1+ 1199022

1)

(45)

Combining this with (39) and (41) leads to

1198862

2=

1198613

1(1199012+ 1199022)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(46)

From (44) and (46) it follows that

1198863minus 1205831198862

2= 1198611[(ℎ (120583) +

1

8 (3 minus 2120572))1199012

+(ℎ (120583) minus1

8 (3 minus 2120572)) 1199022]

(47)

where

ℎ (120583) =1198612

1(1 minus 120583)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(48)

Then in view of (8) and (12) we conclude that

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) 0 le

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 le

1

8 (3 minus 2120572)

41198611

1003816100381610038161003816ℎ (120583)1003816100381610038161003816

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 ge

1

8 (3 minus 2120572)

(49)

Taking 120583 = 1 or 120583 = 0 we get the following

Corollary 11 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) (50)

Corollary 12 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381611988631003816100381610038161003816 le

1198611

2 (3 minus 2120572)

1198611minus 1198612

1198612

1

isin (minusinfin120572 minus 3

2 (2 minus 120572)]

cup [120572 (1 minus 120572)

2 (2 minus 120572)2infin)

1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1198611minus 1198612

1198612

1

isin [120572 minus 3

2 (2 minus 120572)3120572 minus 120572

2

minus 3

2 (2 minus 120572)2)

cup (3120572 minus 120572

2

minus 3

2 (2 minus 120572)2120572 (1 minus 120572)

2 (2 minus 120572)2]

(51)

Corollary 13 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(52)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

3 minus 2120572 (53)

Corollary 14 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(54)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

3 minus 3120572 + 1205722 (55)

Remark 15 Corollaries 13 and 14 provide an improvement ofthe estimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

Conflict of Interests

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

References

[1] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[2] M Lewin ldquoOn a coefficient problem for bi-univalent functionsrdquoProceedings of the American Mathematical Society vol 18 pp63ndash68 1967

[3] D A Brannan and J G Clunie ldquoAspects of comtemporarycomplex analysisrdquo in Proceedings of the NATO Advanced StudyInstute Held at University of Durham July 1ndash20 1979 AcademicPress New York NY USA 1980

[4] E Netanyahu ldquoThe minimal distance of the image boundaryfrom the orijin and the second coefficient of a univalentfunction in |119911| lt 1rdquo Archive for Rational Mechanics andAnalysis vol 32 pp 100ndash112 1969

[5] D A Brannan and T S Taha ldquoOn some classes of bi-univalentfunctionsrdquo Studia Universitatis Babes-Bolyai Mathematica vol31 no 2 pp 70ndash77 1986

[6] S Altinkaya and S Yalcin ldquoInitial coefficient bounds for ageneral class of biunivalent functionsrdquo International Journal ofAnalysis vol 2014 Article ID 867871 4 pages 2014

[7] O Crisan ldquoCoefficient estimates for certain subclasses of bi-univalent functionsrdquo General Mathematics Notes vol 16 no 2pp 93ndash102 2013

[8] B A Frasin and M K Aouf ldquoNew subclasses of bi-univalentfunctionsrdquoAppliedMathematics Letters vol 24 no 9 pp 1569ndash1573 2011

[9] B S Keerthi and B Raja ldquoCoefficient inequality for certainnew subclasses of analytic bi-univalent functionsrdquo TheoreticalMathematics amp Applications vol 3 no 1 pp 1ndash10 2013

[10] N Magesh and J Yamini ldquoCoefficient bounds for certain sub-classes of bi-univalent functionsrdquo International MathematicalForum Journal for Theory and Applications vol 8 no 25-28pp 1337ndash1344 2013

6 International Scholarly Research Notices

[11] H M Srivastava A K Mishra and P Gochhayat ldquoCertainsubclasses of analytic and bi-univalent functionsrdquo AppliedMathematics Letters vol 23 no 10 pp 1188ndash1192 2010

[12] Q H Xu Y C Gui andHM Srivastava ldquoCoefficient estimatesfor a certain subclass of analytic and bi-univalent functionsrdquoApplied Mathematics Letters vol 25 no 6 pp 990ndash994 2012

[13] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis (Tianjin 1992) Conference on Proceed-ings Lecture Notes for Analysis I pp 157ndash169 InternationalPress Cambridge Mass USA 1994

[14] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[15] R M Ali S K Lee V Ravichandran and S SupramanianldquoCoefficient estimates for bi-univalent Ma-Minda starlike andconvex functionsrdquo Applied Mathematics Letters vol 25 no 3pp 344ndash351 2012

[16] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[17] Z G Wang C Y Gao and S M Yuan ldquoOn certain subclassesof close-to-convex and quasi-convex functions with respectto k-symmetric pointsrdquo Journal of Mathematical Analysis andApplications vol 322 no 1 pp 97ndash106 2006

[18] V Ravichandran ldquoStarlike and convex functions with respectto conjugate pointsrdquo Acta Mathematica vol 20 no 1 pp 31ndash372004

[19] P Zaprawa ldquoOn the Fekete-Szego problem for classes ofbi-univalent functionsrdquo Bulletin of the Belgian MathematicalSociety Simon Stevin vol 21 no 1 pp 169ndash178 2014

[20] C Pommerenke Univalent Functions Vandenhoeck ampRuprecht Gottingen Germany 1975

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

International Scholarly Research Notices 5

By adding (40) to (42) we get

4 (3 minus 3120572 + 1205722

) 1198862

2=1

21198611(1199012+ 1199022) minus

1

4(1198611minus 1198612) (1199012

1+ 1199022

1)

(45)

Combining this with (39) and (41) leads to

1198862

2=

1198613

1(1199012+ 1199022)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(46)

From (44) and (46) it follows that

1198863minus 1205831198862

2= 1198611[(ℎ (120583) +

1

8 (3 minus 2120572))1199012

+(ℎ (120583) minus1

8 (3 minus 2120572)) 1199022]

(47)

where

ℎ (120583) =1198612

1(1 minus 120583)

8 [(3 minus 3120572 + 1205722) 1198612

1+ 2 (2 minus 120572)

2

(1198611minus 1198612)]

(48)

Then in view of (8) and (12) we conclude that

100381610038161003816100381610038161198863minus 1205831198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) 0 le

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 le

1

8 (3 minus 2120572)

41198611

1003816100381610038161003816ℎ (120583)1003816100381610038161003816

1003816100381610038161003816ℎ (120583)1003816100381610038161003816 ge

1

8 (3 minus 2120572)

(49)

Taking 120583 = 1 or 120583 = 0 we get the following

Corollary 11 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

1198611

2 (3 minus 2120572) (50)

Corollary 12 If 119891 isin m119878Σ(120572 120601) then

100381610038161003816100381611988631003816100381610038161003816 le

1198611

2 (3 minus 2120572)

1198611minus 1198612

1198612

1

isin (minusinfin120572 minus 3

2 (2 minus 120572)]

cup [120572 (1 minus 120572)

2 (2 minus 120572)2infin)

1198613

1

100381610038161003816100381610038162 (3 minus 3120572 + 1205722) 119861

2

1+ 4 (2 minus 120572)

2

(1198611minus 1198612)10038161003816100381610038161003816

1198611minus 1198612

1198612

1

isin [120572 minus 3

2 (2 minus 120572)3120572 minus 120572

2

minus 3

2 (2 minus 120572)2)

cup (3120572 minus 120572

2

minus 3

2 (2 minus 120572)2120572 (1 minus 120572)

2 (2 minus 120572)2]

(51)

Corollary 13 If

120601 (119911) = (1 + 119911

1 minus 119911)

120573

= 1 + 2120573119911 + 21205732

1199112

+ sdot sdot sdot (0 lt 120573 le 1)

(52)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le

120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

120573

3 minus 2120572 (53)

Corollary 14 If

120601 (119911) =1 + (1 minus 2120573) 119911

1 minus 119911

= 1 + 2 (1 minus 120573) 119911 + 2 (1 minus 120573) 1199112

+ sdot sdot sdot (0 le 120573 lt 1)

(54)

then inequalities (50) and (51) become

100381610038161003816100381610038161198863minus 1198862

2

10038161003816100381610038161003816le1 minus 120573

3 minus 2120572

100381610038161003816100381611988631003816100381610038161003816 le

1 minus 120573

3 minus 3120572 + 1205722 (55)

Remark 15 Corollaries 13 and 14 provide an improvement ofthe estimate 10038161003816100381610038161198863

1003816100381610038161003816 obtained by Crisan [7]

Conflict of Interests

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

References

[1] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[2] M Lewin ldquoOn a coefficient problem for bi-univalent functionsrdquoProceedings of the American Mathematical Society vol 18 pp63ndash68 1967

[3] D A Brannan and J G Clunie ldquoAspects of comtemporarycomplex analysisrdquo in Proceedings of the NATO Advanced StudyInstute Held at University of Durham July 1ndash20 1979 AcademicPress New York NY USA 1980

[4] E Netanyahu ldquoThe minimal distance of the image boundaryfrom the orijin and the second coefficient of a univalentfunction in |119911| lt 1rdquo Archive for Rational Mechanics andAnalysis vol 32 pp 100ndash112 1969

[5] D A Brannan and T S Taha ldquoOn some classes of bi-univalentfunctionsrdquo Studia Universitatis Babes-Bolyai Mathematica vol31 no 2 pp 70ndash77 1986

[6] S Altinkaya and S Yalcin ldquoInitial coefficient bounds for ageneral class of biunivalent functionsrdquo International Journal ofAnalysis vol 2014 Article ID 867871 4 pages 2014

[7] O Crisan ldquoCoefficient estimates for certain subclasses of bi-univalent functionsrdquo General Mathematics Notes vol 16 no 2pp 93ndash102 2013

[8] B A Frasin and M K Aouf ldquoNew subclasses of bi-univalentfunctionsrdquoAppliedMathematics Letters vol 24 no 9 pp 1569ndash1573 2011

[9] B S Keerthi and B Raja ldquoCoefficient inequality for certainnew subclasses of analytic bi-univalent functionsrdquo TheoreticalMathematics amp Applications vol 3 no 1 pp 1ndash10 2013

[10] N Magesh and J Yamini ldquoCoefficient bounds for certain sub-classes of bi-univalent functionsrdquo International MathematicalForum Journal for Theory and Applications vol 8 no 25-28pp 1337ndash1344 2013

6 International Scholarly Research Notices

[11] H M Srivastava A K Mishra and P Gochhayat ldquoCertainsubclasses of analytic and bi-univalent functionsrdquo AppliedMathematics Letters vol 23 no 10 pp 1188ndash1192 2010

[12] Q H Xu Y C Gui andHM Srivastava ldquoCoefficient estimatesfor a certain subclass of analytic and bi-univalent functionsrdquoApplied Mathematics Letters vol 25 no 6 pp 990ndash994 2012

[13] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis (Tianjin 1992) Conference on Proceed-ings Lecture Notes for Analysis I pp 157ndash169 InternationalPress Cambridge Mass USA 1994

[14] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[15] R M Ali S K Lee V Ravichandran and S SupramanianldquoCoefficient estimates for bi-univalent Ma-Minda starlike andconvex functionsrdquo Applied Mathematics Letters vol 25 no 3pp 344ndash351 2012

[16] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[17] Z G Wang C Y Gao and S M Yuan ldquoOn certain subclassesof close-to-convex and quasi-convex functions with respectto k-symmetric pointsrdquo Journal of Mathematical Analysis andApplications vol 322 no 1 pp 97ndash106 2006

[18] V Ravichandran ldquoStarlike and convex functions with respectto conjugate pointsrdquo Acta Mathematica vol 20 no 1 pp 31ndash372004

[19] P Zaprawa ldquoOn the Fekete-Szego problem for classes ofbi-univalent functionsrdquo Bulletin of the Belgian MathematicalSociety Simon Stevin vol 21 no 1 pp 169ndash178 2014

[20] C Pommerenke Univalent Functions Vandenhoeck ampRuprecht Gottingen Germany 1975

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

6 International Scholarly Research Notices

[11] H M Srivastava A K Mishra and P Gochhayat ldquoCertainsubclasses of analytic and bi-univalent functionsrdquo AppliedMathematics Letters vol 23 no 10 pp 1188ndash1192 2010

[12] Q H Xu Y C Gui andHM Srivastava ldquoCoefficient estimatesfor a certain subclass of analytic and bi-univalent functionsrdquoApplied Mathematics Letters vol 25 no 6 pp 990ndash994 2012

[13] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis (Tianjin 1992) Conference on Proceed-ings Lecture Notes for Analysis I pp 157ndash169 InternationalPress Cambridge Mass USA 1994

[14] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[15] R M Ali S K Lee V Ravichandran and S SupramanianldquoCoefficient estimates for bi-univalent Ma-Minda starlike andconvex functionsrdquo Applied Mathematics Letters vol 25 no 3pp 344ndash351 2012

[16] K Sakaguchi ldquoOn a certain univalent mappingrdquo Journal of theMathematical Society of Japan vol 11 pp 72ndash75 1959

[17] Z G Wang C Y Gao and S M Yuan ldquoOn certain subclassesof close-to-convex and quasi-convex functions with respectto k-symmetric pointsrdquo Journal of Mathematical Analysis andApplications vol 322 no 1 pp 97ndash106 2006

[18] V Ravichandran ldquoStarlike and convex functions with respectto conjugate pointsrdquo Acta Mathematica vol 20 no 1 pp 31ndash372004

[19] P Zaprawa ldquoOn the Fekete-Szego problem for classes ofbi-univalent functionsrdquo Bulletin of the Belgian MathematicalSociety Simon Stevin vol 21 no 1 pp 169ndash178 2014

[20] C Pommerenke Univalent Functions Vandenhoeck ampRuprecht Gottingen Germany 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of