Research ArticleFekete-Szegouml Type Coefficient Inequalities for CertainSubclass of Analytic Functions and Their Applications Involvingthe Owa-Srivastava Fractional Operator

Serap Bulut

Civil Aviation College Kocaeli University Arslanbey Campus 41285 Izmit Kocaeli Turkey

Correspondence should be addressed to Serap Bulut bulutserapyahoocom

Received 7 November 2013 Accepted 28 January 2014 Published 13 March 2014

Academic Editor Tohru Ozawa

Copyright copy 2014 Serap Bulut This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new subclass of analytic functions is introduced For this class firstly the Fekete-Szego type coefficient inequalities are derivedVarious known or new special cases of our results are also pointed out Secondly some applications of ourmain results involving theOwa-Srivastava fractional operator are consideredThus as one of these applications of our result we obtain the Fekete-Szego typeinequality for a class of normalized analytic functions which is defined here by means of the Hadamard product (or convolution)and the Owa-Srivastava fractional operator

1 Introduction and Definitions

LetA denote the class of functions of the form

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 (1)

which are analytic in the unit disk

U = 119911 isin C |119911| lt 1 (2)

Also let S denote the subclass of A consisting of univalentfunctions in U

Fekete and Szego [1] proved a noticeable result that theestimate

100381610038161003816100381610038161198863 minus 1205821198862

2

10038161003816100381610038161003816 le

minus4120582 + 3 120582 le 0

1 + 2 exp( minus2120582

1 minus 120582) 0 le 120582 le 1

4120582 minus 3 120582 ge 1

(3)

holds for119891 isin SThe result is sharp in the sense that for each 120582there is a function in the class under consideration for whichequality holds

The coefficient functional

120601120582(119891) = 119886

3minus 12058211988622=1

6(119891101584010158401015840 (0) minus

3120582

2(11989110158401015840 (0))

2

) (4)

on 119891 isin A represents various geometric quantities as well asin the sense that this behaves well with respect to the rotationnamely

120601120582(119890minus119894120579119891 (119890119894120579119911)) = 1198902119894120579120601

120582(119891) (120579 isin R) (5)

In fact rather than the simplest case when

1206010(119891) = 119886

3 (6)

we have several important ones For example

1206011(119891) = 119886

3minus 11988622

(7)

represents 119878119891(0)6 where 119878

119891denotes the Schwarzian deriva-

tive

119878119891(119911) = (

11989110158401015840 (119911)

1198911015840 (119911))

1015840

minus1

2(11989110158401015840 (119911)

1198911015840 (119911))

2

(8)

Moreover the first two nontrivial coefficients of the 119899th roottransform

(119891 (119911119899))1119899

= 119911 + 119888119899+1

119911119899+1 + 1198882119899+1

1199112119899+1 + sdot sdot sdot (9)

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2014 Article ID 490359 8 pageshttpdxdoiorg1011552014490359

2 International Journal of Analysis

of 119891 with the power series (1) are written by

119888119899+1

=1198862

119899

1198882119899+1

=1198863

119899+(119899 minus 1) 1198862

2

21198992

(10)

so that

1198863minus 12058211988622= 119899 (119888

2119899+1minus 1205831198882119899+1

) (11)

where

120583 = 120582119899 +119899 minus 1

2 (12)

Thus it is quite natural to ask about inequalities for120601120582corresponding to subclasses of S This is called Fekete-

Szego problem Actually many authors have considered thisproblem for typical classes of univalent functions (see eg[1ndash12])

For two functions 119891 and 119892 analytic in U we say that thefunction 119891(119911) is subordinate to 119892(119911) in U and we write

119891 (119911) ≺ 119892 (119911) (119911 isin U) (13)

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

119908 (0) = 0 |119908 (119911)| lt 1 (119911 isin U) (14)

such that

119891 (119911) = 119892 (119908 (119911)) (119911 isin U) (15)

In particular if the function 119892 is univalent in U the abovesubordination is equivalent to

119891 (0) = 119892 (0) 119891 (U) sub 119892 (U) (16)

Let 120593(119911) be an analytic function with

120593 (0) = 1 1205931015840 (0) gt 0 Re 120593 (119911) gt 0 (119911 isin U)

(17)

which maps the open unit disk U onto a star-like region withrespect to 1 and is symmetric with respect to the real axis

This paper contains analogues of (3) for the followingclasses of analytic functions

Definition 1 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (18)

A function 119891 isin A is said to be in the class R119887(120572 120574 120593) if it

satisfies the following subordination condition

1 +1

119887((1 minus 120572 + 2120574)

119891 (119911)

119911+ (120572 minus 2120574) 1198911015840 (119911)

+12057411991111989110158401015840 (119911) minus 1) ≺ 120593 (119911)

(19)

where 120593(119911) is defined to be the same as above for 119911 isin U

Remark 2 (i) If we set

120572 = 2120574 + 1 (20)

in Definition 1 then we have the class

R119887(2120574 + 1 120574 120593) = R

119887

120574(120593) (21)

which consists of functions satisfying

1 +1

119887(1198911015840 (119911) + 120574119911119891

10158401015840

(119911) minus 1) ≺ 120593 (119911) (119911 isin U) (22)

This class was introduced by Bansal [13]

(ii) If we set

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (23)

in Definition 1 then we have a new class

R119887(120572 120574

1 + 119860119911

1 + 119861119911) = R

119887(120572 120574 119860 119861) (24)

which consists of functions satisfying

1 +1

119887((1 minus 120572 + 2120574)

119891 (119911)

119911+ (120572 minus 2120574) 1198911015840 (119911)

+ 12057411991111989110158401015840 (119911) minus 1) ≺1 + 119860119911

1 + 119861119911

(25)

Taking

120572 = 2120574 + 1 (26)

in (25) we have the class

R119887(2120574 + 1 120574 119860 119861) = R

119887

120574(119860 119861) (27)

which consists of functions satisfying

1 +1

119887(1198911015840 (119911) + 120574119911119891

10158401015840

(119911) minus 1) ≺1 + 119860119911

1 + 119861119911

(minus1 le 119861 lt 119860 le 1 119911 isin U)

(28)

This class was introduced by Bansal [14]

(iii) If we set

120574 = 0 119887 = 1 119860 = 1 119861 = minus1 (29)

in (25) then we have the class

R1(120572 0 1 minus1) = R (120572) (30)

which consists of functions satisfying

Re(1 minus 120572)119891 (119911)

119911+ 1205721198911015840 (119911) gt 0 (31)

This class was introduced by Murugusundaramoorthy andMagesh [15] The subclass R

1(1 0 1 minus1) = R(1) = R was

studied by MacGregor [16]

International Journal of Analysis 3

We denote by P the class of the analytic functions in U

with

119901 (0) = 1 Re 119901 (119911) gt 0 (32)

We will need the following lemmas

Lemma 3 (see [12]) If 119901 isin P with 119901(119911) = 1+ 1198881119911+ 11988821199112 + sdot sdot sdot

then

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 le

minus4] + 2 ] le 02 0 le ] le 14] minus 2 ] ge 1

(33)

When ] lt 0 or ] gt 1 equality holds true if and only if 119901(119911) is(1+119911)(1minus119911) or one of its rotations If 0 lt ] lt 1 then equalityholds true if and only if 119901(119911) is (1 + 1199112)(1 minus 1199112) or one of itsrotations If ] = 0 then the equality holds true if and only if

119901 (119911) = (1

2+1

2120578)

1 + 119911

1 minus 119911+ (

1

2minus1

2120578)

1 minus 119911

1 + 119911(0 le 120578 le 1)

(34)

or one of its rotations If ] = 1 then the equality holds true ifand only if 119901(119911) is the reciprocal of one of the functions suchthat the equality holds true in the case when ] = 0

Although the above upper bound is sharp in the case when0 lt ] lt 1 it can be further improved as follows

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 + ]1003816100381610038161003816119888110038161003816100381610038162

le 2 (0 lt ] le1

2)

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 + (1 minus ]) 1003816100381610038161003816119888110038161003816100381610038162

le 2 (1

2lt ] le 1)

(35)

Lemma 4 (see [17]) Let 119901 isin P with 119901(119911) = 1 + 1198881119911 + 11988821199112 +

sdot sdot sdot Then for any complex number ]100381610038161003816100381610038161198882 minus ]1198882

1

10038161003816100381610038161003816 le 2max 1 |2] minus 1| (36)

and the result is sharp for the functions given by

119901 (119911) =1 + 1199112

1 minus 1199112 119901 (119911) =

1 + 119911

1 minus 119911 (37)

2 Fekete-Szegouml Problem for the FunctionClass R

119887(120572 120574 120593)

By making use of Lemma 3 we first prove the Fekete-Szegotype inequalities asserted byTheorem 5 below

Theorem 5 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (38)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (39)

where

1198611gt 0 119861

2ge 0 (40)

If 119891(119911) given by (1) belongs to the function class R119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574)(1198612

1198611

minus 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(41)

where

1205901=(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205903=

1198612(1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(42)

If 1205901le 120583 le 120590

3 then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 +(1 + 120572)2

1198611119887 (1 + 2120572 + 2120574)

times [1 minus1198612

1198611

+ 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2]1003816100381610038161003816119886210038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(43)

Furthermore if 1205903le 120583 le 120590

2 then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 +(1 + 120572)2

1198611119887 (1 + 2120572 + 2120574)

times [1 +1198612

1198611

minus 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2]1003816100381610038161003816119886210038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(44)

Each of these results is sharp

Proof Since 119891 isin R119887(120572 120574 120593) we have

ℎ (119911) ≺ 120593 (119911) (45)

where

ℎ (119911) = 1 +1

119887((1 minus 120572 + 2120574)

119891 (119911)

119911+ (120572 minus 2120574) 1198911015840 (119911)

+12057411991111989110158401015840 (119911) minus 1)

= 1 + ℎ1119911 + ℎ21199112 + sdot sdot sdot

(46)

4 International Journal of Analysis

From (46) we have

ℎ1=1

119887(1 + 120572) 119886

2 ℎ

2=1

119887(1 + 2120572 + 2120574) 119886

3 (47)

Since 120593(119911) is univalent and ℎ(119911) ≺ 120593(119911) the function

1199011(119911) =

1 + 120593minus1 (ℎ (119911))

1 minus 120593minus1 (ℎ (119911))= 1 + 119888

1119911 + 11988821199112 + 119888

31199113 + sdot sdot sdot

(48)

is analytic and has a positive real part in U Also we have

ℎ (119911) = 120593(1199011(119911) minus 1

1199011(119911) + 1

)

= 1 +11986111198881

2119911 + [

1198611

2(1198882minus11988821

2) +

119861211988821

4] 1199112 + sdot sdot sdot

(49)

Thus by (47) and (49) we get

1198862=

11986111198881119887

2 (1 + 120572)

1198863=

119887

(1 + 2120572 + 2120574)[1198611

2(1198882minus11988821

2) +

119861211988821

4]

(50)

Taking into account (50) we obtain

1198863minus 12058311988622=

119887

(1 + 2120572 + 2120574)

times [1198611

2(1198882minus11988821

2) +

119861211988821

4] minus 120583

11986121119888211198872

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)(1198882minus 12057511988821)

(51)

where

120575 =1

2(1 minus

1198612

1198611

+1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2) (52)

The assertion of Theorem 5 now follows by an application ofLemma 3 On the other hand using (51) for the values of 120590

1le

120583 le 1205903 we have100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816 + (120583 minus 1205901)1003816100381610038161003816119886210038161003816100381610038162

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (120583 minus 1205901)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (120583 minus(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)100381610038161003816100381610038161198882 minus 120575119888

2

1

10038161003816100381610038161003816 + 1205751003816100381610038161003816119888110038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(53)

Similarly for the values of 1205903le 120583 le 120590

2 we get

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 + (1205902 minus 120583)1003816100381610038161003816119886210038161003816100381610038162

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (1205902minus 120583)

1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

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

11986121119887 (1 + 2120572 + 2120574)

minus 120583)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)100381610038161003816100381610038161198882 minus 120575119888

2

1

10038161003816100381610038161003816 + (1 minus 120575)1003816100381610038161003816119888110038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(54)

To show that the bounds asserted byTheorem 5 are sharp wedefine the following functions

119870120593119899

(119911) (119899 = 2 3 ) (55)

with119870120593119899

(0) = 0 = 1198701015840

120593119899

(0) minus 1 (56)

by

1 +1

119887((1 minus 120572 + 2120574)

119870120593119899

(119911)

119911+ (120572 minus 2120574)1198701015840

120593119899

(119911)

+ 12057411991111987010158401015840120593119899

(119911) minus 1) = 120593 (119911119899minus1)

(57)

International Journal of Analysis 5

and the functions 119865120578(119911) and 119866

120578(119911) (0 le 120578 le 1) with

119865120578(0) = 0 = 119865

1015840

120578(0) minus 1 119866

120578(0) = 0 = 119866

1015840

120578(0) minus 1

(58)

by

1 +1

119887((1 minus 120572 + 2120574)

119865120578(119911)

119911+ (120572 minus 2120574) 1198651015840

120578(119911)

+12057411991111986510158401015840120578(119911) minus 1) = 120593(

119911 (119911 + 120578)

1 + 120578119911)

1 +1

119887((1 minus 120572 + 2120574)

119866120578(119911)

119911+ (120572 minus 2120574)1198661015840

120578(119911)

+ 12057411991111986610158401015840120578(119911) minus 1) = 120593(minus

119911 (119911 + 120578)

1 + 120578119911)

(59)

respectively Then clearly the functions 119870120593119899

119865120578 and 119866

120578isin

R119887(120572 120574 120593) We also write

119870120593= 1198701205932

(60)

If 120583 lt 1205901or 120583 gt 120590

2 then the equality inTheorem 5 holds true

if and only if119891 is119870120593or one of its rotationsWhen120590

1lt 120583 lt 120590

2

then the equality holds true if and only if119891 is1198701205933

or one of itsrotations If 120583 = 120590

1 then the equality holds true if and only

if 119891 is 119865120578or one of its rotations If 120583 = 120590

2 then the equality

holds true if and only if 119891 is 119866120578or one of its rotations

By making use of Lemma 4 we immediately obtain thefollowing Fekete-Szego type inequality

Theorem 6 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (61)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (62)

where

1198611gt 0 119861

2ge 0 (63)

If 119891(119911) given by (1) belongs to the function class R119887(120572 120574 120593)

then for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

(1 + 2120572 + 2120574)max1

100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(64)

The result is sharp

Remark 7 The coefficient bounds for |1198862| and |119886

3| are special

cases of those asserted byTheorem 5Taking 120572 = 2120574 + 1 in Theorem 6 we have the following

corollary

Corollary 8 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (65)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (66)

where

1198611gt 0 119861

2ge 0 (67)

If 119891(119911) given by (1) belongs to the function class R119887120574(120593) then

for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

3 (1 + 2120574)max1

1003816100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus31205831198611119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(68)

The result is sharp

If we set

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (69)

in Theorem 6 then we have

1198611= 119860 minus 119861 119861

2= minus119861 (119860 minus 119861) (70)

So we get the following corollary

Corollary 9 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (71)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (72)

If 119891(119911) given by (1) belongs to the function classR119887(120572 120574 119860 119861)

then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

(1 + 2120572 + 2120574)

timesmax1100381610038161003816100381610038161003816100381610038161003816119861 +

120583 (119860 minus 119861) 119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(73)

The result is sharp

Putting 120572 = 2120574+1 in Corollary 9 we obtain the followingcorollary

Corollary 10 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (74)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (75)

6 International Journal of Analysis

If119891(119911) given by (1) belongs to the function classR119887120574(119860 119861) then

for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

3 (1 + 2120574)

timesmax11003816100381610038161003816100381610038161003816100381610038161003816119861 +

3120583 (119860 minus 119861) 119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(76)

The result is sharp

Also putting 120574 = 0 119887 = 1 119860 = 1 and 119861 = minus1 inCorollary 9 we obtain the following corollary

Corollary 11 Let 120572 gt 0 If 119891(119911) given by (1) belongs to thefunction classR(120572) then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le2

(1 + 2120572)max1

1003816100381610038161003816100381610038161003816100381610038161 minus

2120583 (1 + 2120572)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816 (77)

3 Applications to Analytic FunctionsDefined by Using Fractional CalculusOperators and Convolution

The subject of fractional calculus (ie calculus of integralsand derivatives of any arbitrary real or complex order) hasgained considerable popularity and importance during thepast three decades or so For the applications of the resultsgiven in the preceding sections we first introduce the classR120588

119887(120572 120574 120593) which is defined by means of the Hadamard

product (or convolution) and a certain operator of fractionalcalculus known as the Owa-Srivastava operator (see eg[18 19])

Definition 12 The fractional integral of order 120588 is defined fora function 119891(119911) by

119863minus120588119911119891 (119911) =

1

Γ (120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)1minus120588

119889120577 (120588 gt 0) (78)

where the function 119891(119911) is analytic in a simply connecteddomain of the complex 119911-plane containing the origin and themultiplicity of (119911minus120577)120588minus1 is removed by requiring log(119911minus120577) tobe real when 119911 minus 120577 gt 0

Definition 13 The fractional derivative of order 120588 is definedfor a function 119891(119911) by

119863120588119911119891 (119911) =

1

Γ (1 minus 120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)120588119889120577 (0 le 120588 lt 1)

(79)

where 119891(119911) is constrained and the multiplicity of (119911 minus 120577)minus120588 isremoved as in Definition 12

Definition 14 Under the hypotheses of Definition 13 thefractional derivative of order 119899 + 120588 is defined for a function119891(119911) by

119863119899+120588119911

119891 (119911) =119889119899

119889119911119899(119863120588119911119891 (119911))

(0 le 120588 lt 1 119899 isin N0= N cup 0)

(80)

UsingDefinitions 12 13 and 14 fractional derivatives andfractional integrals Owa and Srivastava [20] introduced theoperatorΩ120588 A rarr A defined by

(Ω120588119891) (119911) = Γ (2 minus 120588) 119911120588119863120588119911119891 (119911) 120588 = 2 3 4 (81)

= 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119886119899119911119899 (82)

This operator is known as the Owa-Srivastava operator Interms of the Owa-Srivastava operator Ω120588 defined by (81) wenow introduce the function classR120588

119887(120572 120574 120593) in the following

way

R120588

119887(120572 120574 120593) = 119891 isin A Ω120588119891 isin R

119887(120572 120574 120593) (83)

Note that the function classR120588119887(120572 120574 120593) is a special case of the

function classR119892119887(120572 120574 120593) when

119892 (119911) = 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119911119899 (84)

Now suppose that

119892 (119911) = 119911 +infin

sum119899=2

119892119899119911119899 (119892

119899gt 0) (85)

Since

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 isin R

119892

119887(120572 120574 120593)

lArrrArr (119891 lowast 119892) (119911) = 119911 +infin

sum119899=2

119892119899119886119899119911119899 isin R

119887(120572 120574 120593)

(86)

we can obtain the coefficient estimates for functions inthe class R119892

119887(120572 120574 120593) from the corresponding estimates for

functions in the class R119887(120572 120574 120593) By applying Theorem 5 to

the following Hadamard product (or convolution)

(119891 lowast 119892) (119911) = 119911 + 119892211988621199112 + 119892

311988631199113 + sdot sdot sdot (87)

we get the following theorem after an obvious change of theparameter 120583

Theorem 15 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (88)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (89)

International Journal of Analysis 7

where

1198611gt 0 119861

2ge 0 119892

119899gt 0 (119899 = 3 4 ) (90)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574) 1198923

(1198612

1198611

minus 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574) 1198923

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574) 1198923

(minus1198612

1198611

+ 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(91)

where

1205901=11989222

1198923

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=11989222

1198923

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(92)

Each of these results is sharp

When 119892 corresponds to the Owa-Srivastava operatorgiven in (82) we obtain

1198922=Γ (3) Γ (2 minus 120588)

Γ (3 minus 120588)=

2

2 minus 120588 (93)

1198923=Γ (4) Γ (2 minus 120588)

Γ (4 minus 120588)=

6

(2 minus 120588) (3 minus 120588) (94)

For 1198922and 1198923given by (93) and (94) respectivelyTheorem 15

reduces to the following result

Theorem 16 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (95)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (96)

where

1198611gt 0 119861

2ge 0 (97)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(1198612

1198611

minus 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(98)

where

1205901=2 (3 minus 120588)

3 (2 minus 120588)

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=2 (3 minus 120588)

3 (2 minus 120588)

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(99)

Each of these results is sharp

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] M Fekete and G Szego ldquoEine bemerkung uber ungeradeschlichte funktionenrdquo Journal of the London MathematicalSociety vol 8 pp 85ndash89 1933

[2] H R Abdel-Gawad and D K Thomas ldquoThe Fekete-Szegoproblem for strongly close-to-convex functionsrdquo Proceedings ofthe American Mathematical Society vol 114 no 2 pp 345ndash3491992

[3] H S Al-Amiri ldquoCertain generalizations of prestarlike func-tionsrdquoAustralianMathematical Society A vol 28 no 3 pp 325ndash334 1979

[4] J H Choi Y C Kim and T Sugawa ldquoA general approach to theFekete-Szego problemrdquo Journal of the Mathematical Society ofJapan vol 59 no 3 pp 707ndash727 2007

[5] A Chonweerayoot D K Thomas and W UpakarnitikasetldquoOn the Fekete-Szego theorem for close-to-convex functionsrdquoInstitut Mathematique vol 66 pp 18ndash26 1992

[6] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquo Mathematica Japonica vol 44 no3 pp 507ndash511 1996

[7] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquoMathematica Japonica vol 47 no 1pp 125ndash132 1998

[8] S Kanas and A Lecko ldquoOn the Fekete-Szego problem and thedomain of convexity for a certain class of univalent functionsrdquoFolia ScientiarumUniversitatis Technicae Resoviensis no 10 pp49ndash57 1990

8 International Journal of Analysis

[9] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[10] W Koepf ldquoOn the Fekete-Szego problem for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 101 no 1 pp 89ndash95 1987

[11] R R London ldquoFekete-Szego inequalities for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 117 no 4 pp 947ndash950 1993

[12] 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

[13] D Bansal ldquoFekete-Szego problem for a new class of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2011 Article ID 143095 5 pages 2011

[14] D Bansal ldquoUpper bound of second Hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[15] G Murugusundaramoorthy and N Magesh ldquoCoefficientinequalities for certain classes of analytic functions associatedwith Hankel determinantrdquo Bulletin of Mathematical Analysisand Applications vol 1 no 3 pp 85ndash89 2009

[16] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962

[17] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences (FJMS) vol 15 no 2 pp171ndash180 2004

[18] S Owa ldquoOn the distortion theorems Irdquo Kyungpook Mathemat-ical Journal vol 18 no 1 pp 53ndash59 1978

[19] H M Srivastava and S Owa Univalent Functions FractionalCalculus and Their Applications Halsted Press (Ellis HorwoodLimited Chichester) John Wiley and Sons New York NYUSA 1989

[20] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987

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

2 International Journal of Analysis

of 119891 with the power series (1) are written by

119888119899+1

=1198862

119899

1198882119899+1

=1198863

119899+(119899 minus 1) 1198862

2

21198992

(10)

so that

1198863minus 12058211988622= 119899 (119888

2119899+1minus 1205831198882119899+1

) (11)

where

120583 = 120582119899 +119899 minus 1

2 (12)

Thus it is quite natural to ask about inequalities for120601120582corresponding to subclasses of S This is called Fekete-

Szego problem Actually many authors have considered thisproblem for typical classes of univalent functions (see eg[1ndash12])

For two functions 119891 and 119892 analytic in U we say that thefunction 119891(119911) is subordinate to 119892(119911) in U and we write

119891 (119911) ≺ 119892 (119911) (119911 isin U) (13)

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

119908 (0) = 0 |119908 (119911)| lt 1 (119911 isin U) (14)

such that

119891 (119911) = 119892 (119908 (119911)) (119911 isin U) (15)

In particular if the function 119892 is univalent in U the abovesubordination is equivalent to

119891 (0) = 119892 (0) 119891 (U) sub 119892 (U) (16)

Let 120593(119911) be an analytic function with

120593 (0) = 1 1205931015840 (0) gt 0 Re 120593 (119911) gt 0 (119911 isin U)

(17)

which maps the open unit disk U onto a star-like region withrespect to 1 and is symmetric with respect to the real axis

This paper contains analogues of (3) for the followingclasses of analytic functions

Definition 1 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (18)

A function 119891 isin A is said to be in the class R119887(120572 120574 120593) if it

satisfies the following subordination condition

1 +1

119887((1 minus 120572 + 2120574)

119891 (119911)

119911+ (120572 minus 2120574) 1198911015840 (119911)

+12057411991111989110158401015840 (119911) minus 1) ≺ 120593 (119911)

(19)

where 120593(119911) is defined to be the same as above for 119911 isin U

Remark 2 (i) If we set

120572 = 2120574 + 1 (20)

in Definition 1 then we have the class

R119887(2120574 + 1 120574 120593) = R

119887

120574(120593) (21)

which consists of functions satisfying

1 +1

119887(1198911015840 (119911) + 120574119911119891

10158401015840

(119911) minus 1) ≺ 120593 (119911) (119911 isin U) (22)

This class was introduced by Bansal [13]

(ii) If we set

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (23)

in Definition 1 then we have a new class

R119887(120572 120574

1 + 119860119911

1 + 119861119911) = R

119887(120572 120574 119860 119861) (24)

which consists of functions satisfying

1 +1

119887((1 minus 120572 + 2120574)

119891 (119911)

119911+ (120572 minus 2120574) 1198911015840 (119911)

+ 12057411991111989110158401015840 (119911) minus 1) ≺1 + 119860119911

1 + 119861119911

(25)

Taking

120572 = 2120574 + 1 (26)

in (25) we have the class

R119887(2120574 + 1 120574 119860 119861) = R

119887

120574(119860 119861) (27)

which consists of functions satisfying

1 +1

119887(1198911015840 (119911) + 120574119911119891

10158401015840

(119911) minus 1) ≺1 + 119860119911

1 + 119861119911

(minus1 le 119861 lt 119860 le 1 119911 isin U)

(28)

This class was introduced by Bansal [14]

(iii) If we set

120574 = 0 119887 = 1 119860 = 1 119861 = minus1 (29)

in (25) then we have the class

R1(120572 0 1 minus1) = R (120572) (30)

which consists of functions satisfying

Re(1 minus 120572)119891 (119911)

119911+ 1205721198911015840 (119911) gt 0 (31)

This class was introduced by Murugusundaramoorthy andMagesh [15] The subclass R

1(1 0 1 minus1) = R(1) = R was

studied by MacGregor [16]

International Journal of Analysis 3

We denote by P the class of the analytic functions in U

with

119901 (0) = 1 Re 119901 (119911) gt 0 (32)

We will need the following lemmas

Lemma 3 (see [12]) If 119901 isin P with 119901(119911) = 1+ 1198881119911+ 11988821199112 + sdot sdot sdot

then

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 le

minus4] + 2 ] le 02 0 le ] le 14] minus 2 ] ge 1

(33)

When ] lt 0 or ] gt 1 equality holds true if and only if 119901(119911) is(1+119911)(1minus119911) or one of its rotations If 0 lt ] lt 1 then equalityholds true if and only if 119901(119911) is (1 + 1199112)(1 minus 1199112) or one of itsrotations If ] = 0 then the equality holds true if and only if

119901 (119911) = (1

2+1

2120578)

1 + 119911

1 minus 119911+ (

1

2minus1

2120578)

1 minus 119911

1 + 119911(0 le 120578 le 1)

(34)

or one of its rotations If ] = 1 then the equality holds true ifand only if 119901(119911) is the reciprocal of one of the functions suchthat the equality holds true in the case when ] = 0

Although the above upper bound is sharp in the case when0 lt ] lt 1 it can be further improved as follows

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 + ]1003816100381610038161003816119888110038161003816100381610038162

le 2 (0 lt ] le1

2)

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 + (1 minus ]) 1003816100381610038161003816119888110038161003816100381610038162

le 2 (1

2lt ] le 1)

(35)

Lemma 4 (see [17]) Let 119901 isin P with 119901(119911) = 1 + 1198881119911 + 11988821199112 +

sdot sdot sdot Then for any complex number ]100381610038161003816100381610038161198882 minus ]1198882

1

10038161003816100381610038161003816 le 2max 1 |2] minus 1| (36)

and the result is sharp for the functions given by

119901 (119911) =1 + 1199112

1 minus 1199112 119901 (119911) =

1 + 119911

1 minus 119911 (37)

2 Fekete-Szegouml Problem for the FunctionClass R

119887(120572 120574 120593)

By making use of Lemma 3 we first prove the Fekete-Szegotype inequalities asserted byTheorem 5 below

Theorem 5 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (38)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (39)

where

1198611gt 0 119861

2ge 0 (40)

If 119891(119911) given by (1) belongs to the function class R119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574)(1198612

1198611

minus 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(41)

where

1205901=(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205903=

1198612(1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(42)

If 1205901le 120583 le 120590

3 then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 +(1 + 120572)2

1198611119887 (1 + 2120572 + 2120574)

times [1 minus1198612

1198611

+ 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2]1003816100381610038161003816119886210038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(43)

Furthermore if 1205903le 120583 le 120590

2 then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 +(1 + 120572)2

1198611119887 (1 + 2120572 + 2120574)

times [1 +1198612

1198611

minus 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2]1003816100381610038161003816119886210038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(44)

Each of these results is sharp

Proof Since 119891 isin R119887(120572 120574 120593) we have

ℎ (119911) ≺ 120593 (119911) (45)

where

ℎ (119911) = 1 +1

119887((1 minus 120572 + 2120574)

119891 (119911)

119911+ (120572 minus 2120574) 1198911015840 (119911)

+12057411991111989110158401015840 (119911) minus 1)

= 1 + ℎ1119911 + ℎ21199112 + sdot sdot sdot

(46)

4 International Journal of Analysis

From (46) we have

ℎ1=1

119887(1 + 120572) 119886

2 ℎ

2=1

119887(1 + 2120572 + 2120574) 119886

3 (47)

Since 120593(119911) is univalent and ℎ(119911) ≺ 120593(119911) the function

1199011(119911) =

1 + 120593minus1 (ℎ (119911))

1 minus 120593minus1 (ℎ (119911))= 1 + 119888

1119911 + 11988821199112 + 119888

31199113 + sdot sdot sdot

(48)

is analytic and has a positive real part in U Also we have

ℎ (119911) = 120593(1199011(119911) minus 1

1199011(119911) + 1

)

= 1 +11986111198881

2119911 + [

1198611

2(1198882minus11988821

2) +

119861211988821

4] 1199112 + sdot sdot sdot

(49)

Thus by (47) and (49) we get

1198862=

11986111198881119887

2 (1 + 120572)

1198863=

119887

(1 + 2120572 + 2120574)[1198611

2(1198882minus11988821

2) +

119861211988821

4]

(50)

Taking into account (50) we obtain

1198863minus 12058311988622=

119887

(1 + 2120572 + 2120574)

times [1198611

2(1198882minus11988821

2) +

119861211988821

4] minus 120583

11986121119888211198872

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)(1198882minus 12057511988821)

(51)

where

120575 =1

2(1 minus

1198612

1198611

+1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2) (52)

The assertion of Theorem 5 now follows by an application ofLemma 3 On the other hand using (51) for the values of 120590

1le

120583 le 1205903 we have100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816 + (120583 minus 1205901)1003816100381610038161003816119886210038161003816100381610038162

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (120583 minus 1205901)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (120583 minus(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)100381610038161003816100381610038161198882 minus 120575119888

2

1

10038161003816100381610038161003816 + 1205751003816100381610038161003816119888110038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(53)

Similarly for the values of 1205903le 120583 le 120590

2 we get

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 + (1205902 minus 120583)1003816100381610038161003816119886210038161003816100381610038162

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (1205902minus 120583)

1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

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

11986121119887 (1 + 2120572 + 2120574)

minus 120583)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)100381610038161003816100381610038161198882 minus 120575119888

2

1

10038161003816100381610038161003816 + (1 minus 120575)1003816100381610038161003816119888110038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(54)

To show that the bounds asserted byTheorem 5 are sharp wedefine the following functions

119870120593119899

(119911) (119899 = 2 3 ) (55)

with119870120593119899

(0) = 0 = 1198701015840

120593119899

(0) minus 1 (56)

by

1 +1

119887((1 minus 120572 + 2120574)

119870120593119899

(119911)

119911+ (120572 minus 2120574)1198701015840

120593119899

(119911)

+ 12057411991111987010158401015840120593119899

(119911) minus 1) = 120593 (119911119899minus1)

(57)

International Journal of Analysis 5

and the functions 119865120578(119911) and 119866

120578(119911) (0 le 120578 le 1) with

119865120578(0) = 0 = 119865

1015840

120578(0) minus 1 119866

120578(0) = 0 = 119866

1015840

120578(0) minus 1

(58)

by

1 +1

119887((1 minus 120572 + 2120574)

119865120578(119911)

119911+ (120572 minus 2120574) 1198651015840

120578(119911)

+12057411991111986510158401015840120578(119911) minus 1) = 120593(

119911 (119911 + 120578)

1 + 120578119911)

1 +1

119887((1 minus 120572 + 2120574)

119866120578(119911)

119911+ (120572 minus 2120574)1198661015840

120578(119911)

+ 12057411991111986610158401015840120578(119911) minus 1) = 120593(minus

119911 (119911 + 120578)

1 + 120578119911)

(59)

respectively Then clearly the functions 119870120593119899

119865120578 and 119866

120578isin

R119887(120572 120574 120593) We also write

119870120593= 1198701205932

(60)

If 120583 lt 1205901or 120583 gt 120590

2 then the equality inTheorem 5 holds true

if and only if119891 is119870120593or one of its rotationsWhen120590

1lt 120583 lt 120590

2

then the equality holds true if and only if119891 is1198701205933

or one of itsrotations If 120583 = 120590

1 then the equality holds true if and only

if 119891 is 119865120578or one of its rotations If 120583 = 120590

2 then the equality

holds true if and only if 119891 is 119866120578or one of its rotations

By making use of Lemma 4 we immediately obtain thefollowing Fekete-Szego type inequality

Theorem 6 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (61)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (62)

where

1198611gt 0 119861

2ge 0 (63)

If 119891(119911) given by (1) belongs to the function class R119887(120572 120574 120593)

then for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

(1 + 2120572 + 2120574)max1

100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(64)

The result is sharp

Remark 7 The coefficient bounds for |1198862| and |119886

3| are special

cases of those asserted byTheorem 5Taking 120572 = 2120574 + 1 in Theorem 6 we have the following

corollary

Corollary 8 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (65)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (66)

where

1198611gt 0 119861

2ge 0 (67)

If 119891(119911) given by (1) belongs to the function class R119887120574(120593) then

for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

3 (1 + 2120574)max1

1003816100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus31205831198611119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(68)

The result is sharp

If we set

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (69)

in Theorem 6 then we have

1198611= 119860 minus 119861 119861

2= minus119861 (119860 minus 119861) (70)

So we get the following corollary

Corollary 9 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (71)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (72)

If 119891(119911) given by (1) belongs to the function classR119887(120572 120574 119860 119861)

then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

(1 + 2120572 + 2120574)

timesmax1100381610038161003816100381610038161003816100381610038161003816119861 +

120583 (119860 minus 119861) 119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(73)

The result is sharp

Putting 120572 = 2120574+1 in Corollary 9 we obtain the followingcorollary

Corollary 10 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (74)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (75)

6 International Journal of Analysis

If119891(119911) given by (1) belongs to the function classR119887120574(119860 119861) then

for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

3 (1 + 2120574)

timesmax11003816100381610038161003816100381610038161003816100381610038161003816119861 +

3120583 (119860 minus 119861) 119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(76)

The result is sharp

Also putting 120574 = 0 119887 = 1 119860 = 1 and 119861 = minus1 inCorollary 9 we obtain the following corollary

Corollary 11 Let 120572 gt 0 If 119891(119911) given by (1) belongs to thefunction classR(120572) then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le2

(1 + 2120572)max1

1003816100381610038161003816100381610038161003816100381610038161 minus

2120583 (1 + 2120572)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816 (77)

3 Applications to Analytic FunctionsDefined by Using Fractional CalculusOperators and Convolution

The subject of fractional calculus (ie calculus of integralsand derivatives of any arbitrary real or complex order) hasgained considerable popularity and importance during thepast three decades or so For the applications of the resultsgiven in the preceding sections we first introduce the classR120588

119887(120572 120574 120593) which is defined by means of the Hadamard

product (or convolution) and a certain operator of fractionalcalculus known as the Owa-Srivastava operator (see eg[18 19])

Definition 12 The fractional integral of order 120588 is defined fora function 119891(119911) by

119863minus120588119911119891 (119911) =

1

Γ (120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)1minus120588

119889120577 (120588 gt 0) (78)

where the function 119891(119911) is analytic in a simply connecteddomain of the complex 119911-plane containing the origin and themultiplicity of (119911minus120577)120588minus1 is removed by requiring log(119911minus120577) tobe real when 119911 minus 120577 gt 0

Definition 13 The fractional derivative of order 120588 is definedfor a function 119891(119911) by

119863120588119911119891 (119911) =

1

Γ (1 minus 120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)120588119889120577 (0 le 120588 lt 1)

(79)

where 119891(119911) is constrained and the multiplicity of (119911 minus 120577)minus120588 isremoved as in Definition 12

Definition 14 Under the hypotheses of Definition 13 thefractional derivative of order 119899 + 120588 is defined for a function119891(119911) by

119863119899+120588119911

119891 (119911) =119889119899

119889119911119899(119863120588119911119891 (119911))

(0 le 120588 lt 1 119899 isin N0= N cup 0)

(80)

UsingDefinitions 12 13 and 14 fractional derivatives andfractional integrals Owa and Srivastava [20] introduced theoperatorΩ120588 A rarr A defined by

(Ω120588119891) (119911) = Γ (2 minus 120588) 119911120588119863120588119911119891 (119911) 120588 = 2 3 4 (81)

= 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119886119899119911119899 (82)

This operator is known as the Owa-Srivastava operator Interms of the Owa-Srivastava operator Ω120588 defined by (81) wenow introduce the function classR120588

119887(120572 120574 120593) in the following

way

R120588

119887(120572 120574 120593) = 119891 isin A Ω120588119891 isin R

119887(120572 120574 120593) (83)

Note that the function classR120588119887(120572 120574 120593) is a special case of the

function classR119892119887(120572 120574 120593) when

119892 (119911) = 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119911119899 (84)

Now suppose that

119892 (119911) = 119911 +infin

sum119899=2

119892119899119911119899 (119892

119899gt 0) (85)

Since

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 isin R

119892

119887(120572 120574 120593)

lArrrArr (119891 lowast 119892) (119911) = 119911 +infin

sum119899=2

119892119899119886119899119911119899 isin R

119887(120572 120574 120593)

(86)

we can obtain the coefficient estimates for functions inthe class R119892

119887(120572 120574 120593) from the corresponding estimates for

functions in the class R119887(120572 120574 120593) By applying Theorem 5 to

the following Hadamard product (or convolution)

(119891 lowast 119892) (119911) = 119911 + 119892211988621199112 + 119892

311988631199113 + sdot sdot sdot (87)

we get the following theorem after an obvious change of theparameter 120583

Theorem 15 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (88)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (89)

International Journal of Analysis 7

where

1198611gt 0 119861

2ge 0 119892

119899gt 0 (119899 = 3 4 ) (90)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574) 1198923

(1198612

1198611

minus 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574) 1198923

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574) 1198923

(minus1198612

1198611

+ 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(91)

where

1205901=11989222

1198923

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=11989222

1198923

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(92)

Each of these results is sharp

When 119892 corresponds to the Owa-Srivastava operatorgiven in (82) we obtain

1198922=Γ (3) Γ (2 minus 120588)

Γ (3 minus 120588)=

2

2 minus 120588 (93)

1198923=Γ (4) Γ (2 minus 120588)

Γ (4 minus 120588)=

6

(2 minus 120588) (3 minus 120588) (94)

For 1198922and 1198923given by (93) and (94) respectivelyTheorem 15

reduces to the following result

Theorem 16 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (95)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (96)

where

1198611gt 0 119861

2ge 0 (97)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(1198612

1198611

minus 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(98)

where

1205901=2 (3 minus 120588)

3 (2 minus 120588)

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=2 (3 minus 120588)

3 (2 minus 120588)

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(99)

Each of these results is sharp

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] M Fekete and G Szego ldquoEine bemerkung uber ungeradeschlichte funktionenrdquo Journal of the London MathematicalSociety vol 8 pp 85ndash89 1933

[2] H R Abdel-Gawad and D K Thomas ldquoThe Fekete-Szegoproblem for strongly close-to-convex functionsrdquo Proceedings ofthe American Mathematical Society vol 114 no 2 pp 345ndash3491992

[3] H S Al-Amiri ldquoCertain generalizations of prestarlike func-tionsrdquoAustralianMathematical Society A vol 28 no 3 pp 325ndash334 1979

[4] J H Choi Y C Kim and T Sugawa ldquoA general approach to theFekete-Szego problemrdquo Journal of the Mathematical Society ofJapan vol 59 no 3 pp 707ndash727 2007

[5] A Chonweerayoot D K Thomas and W UpakarnitikasetldquoOn the Fekete-Szego theorem for close-to-convex functionsrdquoInstitut Mathematique vol 66 pp 18ndash26 1992

[6] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquo Mathematica Japonica vol 44 no3 pp 507ndash511 1996

[7] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquoMathematica Japonica vol 47 no 1pp 125ndash132 1998

[8] S Kanas and A Lecko ldquoOn the Fekete-Szego problem and thedomain of convexity for a certain class of univalent functionsrdquoFolia ScientiarumUniversitatis Technicae Resoviensis no 10 pp49ndash57 1990

8 International Journal of Analysis

[9] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[10] W Koepf ldquoOn the Fekete-Szego problem for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 101 no 1 pp 89ndash95 1987

[11] R R London ldquoFekete-Szego inequalities for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 117 no 4 pp 947ndash950 1993

[12] 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

[13] D Bansal ldquoFekete-Szego problem for a new class of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2011 Article ID 143095 5 pages 2011

[14] D Bansal ldquoUpper bound of second Hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[15] G Murugusundaramoorthy and N Magesh ldquoCoefficientinequalities for certain classes of analytic functions associatedwith Hankel determinantrdquo Bulletin of Mathematical Analysisand Applications vol 1 no 3 pp 85ndash89 2009

[16] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962

[17] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences (FJMS) vol 15 no 2 pp171ndash180 2004

[18] S Owa ldquoOn the distortion theorems Irdquo Kyungpook Mathemat-ical Journal vol 18 no 1 pp 53ndash59 1978

[19] H M Srivastava and S Owa Univalent Functions FractionalCalculus and Their Applications Halsted Press (Ellis HorwoodLimited Chichester) John Wiley and Sons New York NYUSA 1989

[20] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987

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 Journal of Analysis 3

We denote by P the class of the analytic functions in U

with

119901 (0) = 1 Re 119901 (119911) gt 0 (32)

We will need the following lemmas

Lemma 3 (see [12]) If 119901 isin P with 119901(119911) = 1+ 1198881119911+ 11988821199112 + sdot sdot sdot

then

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 le

minus4] + 2 ] le 02 0 le ] le 14] minus 2 ] ge 1

(33)

When ] lt 0 or ] gt 1 equality holds true if and only if 119901(119911) is(1+119911)(1minus119911) or one of its rotations If 0 lt ] lt 1 then equalityholds true if and only if 119901(119911) is (1 + 1199112)(1 minus 1199112) or one of itsrotations If ] = 0 then the equality holds true if and only if

119901 (119911) = (1

2+1

2120578)

1 + 119911

1 minus 119911+ (

1

2minus1

2120578)

1 minus 119911

1 + 119911(0 le 120578 le 1)

(34)

or one of its rotations If ] = 1 then the equality holds true ifand only if 119901(119911) is the reciprocal of one of the functions suchthat the equality holds true in the case when ] = 0

Although the above upper bound is sharp in the case when0 lt ] lt 1 it can be further improved as follows

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 + ]1003816100381610038161003816119888110038161003816100381610038162

le 2 (0 lt ] le1

2)

100381610038161003816100381610038161198882 minus ]11988821

10038161003816100381610038161003816 + (1 minus ]) 1003816100381610038161003816119888110038161003816100381610038162

le 2 (1

2lt ] le 1)

(35)

Lemma 4 (see [17]) Let 119901 isin P with 119901(119911) = 1 + 1198881119911 + 11988821199112 +

sdot sdot sdot Then for any complex number ]100381610038161003816100381610038161198882 minus ]1198882

1

10038161003816100381610038161003816 le 2max 1 |2] minus 1| (36)

and the result is sharp for the functions given by

119901 (119911) =1 + 1199112

1 minus 1199112 119901 (119911) =

1 + 119911

1 minus 119911 (37)

2 Fekete-Szegouml Problem for the FunctionClass R

119887(120572 120574 120593)

By making use of Lemma 3 we first prove the Fekete-Szegotype inequalities asserted byTheorem 5 below

Theorem 5 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (38)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (39)

where

1198611gt 0 119861

2ge 0 (40)

If 119891(119911) given by (1) belongs to the function class R119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574)(1198612

1198611

minus 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(41)

where

1205901=(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205903=

1198612(1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(42)

If 1205901le 120583 le 120590

3 then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 +(1 + 120572)2

1198611119887 (1 + 2120572 + 2120574)

times [1 minus1198612

1198611

+ 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2]1003816100381610038161003816119886210038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(43)

Furthermore if 1205903le 120583 le 120590

2 then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 +(1 + 120572)2

1198611119887 (1 + 2120572 + 2120574)

times [1 +1198612

1198611

minus 1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2]1003816100381610038161003816119886210038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(44)

Each of these results is sharp

Proof Since 119891 isin R119887(120572 120574 120593) we have

ℎ (119911) ≺ 120593 (119911) (45)

where

ℎ (119911) = 1 +1

119887((1 minus 120572 + 2120574)

119891 (119911)

119911+ (120572 minus 2120574) 1198911015840 (119911)

+12057411991111989110158401015840 (119911) minus 1)

= 1 + ℎ1119911 + ℎ21199112 + sdot sdot sdot

(46)

4 International Journal of Analysis

From (46) we have

ℎ1=1

119887(1 + 120572) 119886

2 ℎ

2=1

119887(1 + 2120572 + 2120574) 119886

3 (47)

Since 120593(119911) is univalent and ℎ(119911) ≺ 120593(119911) the function

1199011(119911) =

1 + 120593minus1 (ℎ (119911))

1 minus 120593minus1 (ℎ (119911))= 1 + 119888

1119911 + 11988821199112 + 119888

31199113 + sdot sdot sdot

(48)

is analytic and has a positive real part in U Also we have

ℎ (119911) = 120593(1199011(119911) minus 1

1199011(119911) + 1

)

= 1 +11986111198881

2119911 + [

1198611

2(1198882minus11988821

2) +

119861211988821

4] 1199112 + sdot sdot sdot

(49)

Thus by (47) and (49) we get

1198862=

11986111198881119887

2 (1 + 120572)

1198863=

119887

(1 + 2120572 + 2120574)[1198611

2(1198882minus11988821

2) +

119861211988821

4]

(50)

Taking into account (50) we obtain

1198863minus 12058311988622=

119887

(1 + 2120572 + 2120574)

times [1198611

2(1198882minus11988821

2) +

119861211988821

4] minus 120583

11986121119888211198872

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)(1198882minus 12057511988821)

(51)

where

120575 =1

2(1 minus

1198612

1198611

+1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2) (52)

The assertion of Theorem 5 now follows by an application ofLemma 3 On the other hand using (51) for the values of 120590

1le

120583 le 1205903 we have100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816 + (120583 minus 1205901)1003816100381610038161003816119886210038161003816100381610038162

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (120583 minus 1205901)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (120583 minus(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)100381610038161003816100381610038161198882 minus 120575119888

2

1

10038161003816100381610038161003816 + 1205751003816100381610038161003816119888110038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(53)

Similarly for the values of 1205903le 120583 le 120590

2 we get

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 + (1205902 minus 120583)1003816100381610038161003816119886210038161003816100381610038162

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (1205902minus 120583)

1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

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

11986121119887 (1 + 2120572 + 2120574)

minus 120583)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)100381610038161003816100381610038161198882 minus 120575119888

2

1

10038161003816100381610038161003816 + (1 minus 120575)1003816100381610038161003816119888110038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(54)

To show that the bounds asserted byTheorem 5 are sharp wedefine the following functions

119870120593119899

(119911) (119899 = 2 3 ) (55)

with119870120593119899

(0) = 0 = 1198701015840

120593119899

(0) minus 1 (56)

by

1 +1

119887((1 minus 120572 + 2120574)

119870120593119899

(119911)

119911+ (120572 minus 2120574)1198701015840

120593119899

(119911)

+ 12057411991111987010158401015840120593119899

(119911) minus 1) = 120593 (119911119899minus1)

(57)

International Journal of Analysis 5

and the functions 119865120578(119911) and 119866

120578(119911) (0 le 120578 le 1) with

119865120578(0) = 0 = 119865

1015840

120578(0) minus 1 119866

120578(0) = 0 = 119866

1015840

120578(0) minus 1

(58)

by

1 +1

119887((1 minus 120572 + 2120574)

119865120578(119911)

119911+ (120572 minus 2120574) 1198651015840

120578(119911)

+12057411991111986510158401015840120578(119911) minus 1) = 120593(

119911 (119911 + 120578)

1 + 120578119911)

1 +1

119887((1 minus 120572 + 2120574)

119866120578(119911)

119911+ (120572 minus 2120574)1198661015840

120578(119911)

+ 12057411991111986610158401015840120578(119911) minus 1) = 120593(minus

119911 (119911 + 120578)

1 + 120578119911)

(59)

respectively Then clearly the functions 119870120593119899

119865120578 and 119866

120578isin

R119887(120572 120574 120593) We also write

119870120593= 1198701205932

(60)

If 120583 lt 1205901or 120583 gt 120590

2 then the equality inTheorem 5 holds true

if and only if119891 is119870120593or one of its rotationsWhen120590

1lt 120583 lt 120590

2

then the equality holds true if and only if119891 is1198701205933

or one of itsrotations If 120583 = 120590

1 then the equality holds true if and only

if 119891 is 119865120578or one of its rotations If 120583 = 120590

2 then the equality

holds true if and only if 119891 is 119866120578or one of its rotations

By making use of Lemma 4 we immediately obtain thefollowing Fekete-Szego type inequality

Theorem 6 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (61)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (62)

where

1198611gt 0 119861

2ge 0 (63)

If 119891(119911) given by (1) belongs to the function class R119887(120572 120574 120593)

then for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

(1 + 2120572 + 2120574)max1

100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(64)

The result is sharp

Remark 7 The coefficient bounds for |1198862| and |119886

3| are special

cases of those asserted byTheorem 5Taking 120572 = 2120574 + 1 in Theorem 6 we have the following

corollary

Corollary 8 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (65)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (66)

where

1198611gt 0 119861

2ge 0 (67)

If 119891(119911) given by (1) belongs to the function class R119887120574(120593) then

for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

3 (1 + 2120574)max1

1003816100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus31205831198611119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(68)

The result is sharp

If we set

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (69)

in Theorem 6 then we have

1198611= 119860 minus 119861 119861

2= minus119861 (119860 minus 119861) (70)

So we get the following corollary

Corollary 9 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (71)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (72)

If 119891(119911) given by (1) belongs to the function classR119887(120572 120574 119860 119861)

then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

(1 + 2120572 + 2120574)

timesmax1100381610038161003816100381610038161003816100381610038161003816119861 +

120583 (119860 minus 119861) 119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(73)

The result is sharp

Putting 120572 = 2120574+1 in Corollary 9 we obtain the followingcorollary

Corollary 10 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (74)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (75)

6 International Journal of Analysis

If119891(119911) given by (1) belongs to the function classR119887120574(119860 119861) then

for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

3 (1 + 2120574)

timesmax11003816100381610038161003816100381610038161003816100381610038161003816119861 +

3120583 (119860 minus 119861) 119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(76)

The result is sharp

Also putting 120574 = 0 119887 = 1 119860 = 1 and 119861 = minus1 inCorollary 9 we obtain the following corollary

Corollary 11 Let 120572 gt 0 If 119891(119911) given by (1) belongs to thefunction classR(120572) then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le2

(1 + 2120572)max1

1003816100381610038161003816100381610038161003816100381610038161 minus

2120583 (1 + 2120572)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816 (77)

3 Applications to Analytic FunctionsDefined by Using Fractional CalculusOperators and Convolution

The subject of fractional calculus (ie calculus of integralsand derivatives of any arbitrary real or complex order) hasgained considerable popularity and importance during thepast three decades or so For the applications of the resultsgiven in the preceding sections we first introduce the classR120588

119887(120572 120574 120593) which is defined by means of the Hadamard

product (or convolution) and a certain operator of fractionalcalculus known as the Owa-Srivastava operator (see eg[18 19])

Definition 12 The fractional integral of order 120588 is defined fora function 119891(119911) by

119863minus120588119911119891 (119911) =

1

Γ (120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)1minus120588

119889120577 (120588 gt 0) (78)

where the function 119891(119911) is analytic in a simply connecteddomain of the complex 119911-plane containing the origin and themultiplicity of (119911minus120577)120588minus1 is removed by requiring log(119911minus120577) tobe real when 119911 minus 120577 gt 0

Definition 13 The fractional derivative of order 120588 is definedfor a function 119891(119911) by

119863120588119911119891 (119911) =

1

Γ (1 minus 120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)120588119889120577 (0 le 120588 lt 1)

(79)

where 119891(119911) is constrained and the multiplicity of (119911 minus 120577)minus120588 isremoved as in Definition 12

Definition 14 Under the hypotheses of Definition 13 thefractional derivative of order 119899 + 120588 is defined for a function119891(119911) by

119863119899+120588119911

119891 (119911) =119889119899

119889119911119899(119863120588119911119891 (119911))

(0 le 120588 lt 1 119899 isin N0= N cup 0)

(80)

UsingDefinitions 12 13 and 14 fractional derivatives andfractional integrals Owa and Srivastava [20] introduced theoperatorΩ120588 A rarr A defined by

(Ω120588119891) (119911) = Γ (2 minus 120588) 119911120588119863120588119911119891 (119911) 120588 = 2 3 4 (81)

= 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119886119899119911119899 (82)

This operator is known as the Owa-Srivastava operator Interms of the Owa-Srivastava operator Ω120588 defined by (81) wenow introduce the function classR120588

119887(120572 120574 120593) in the following

way

R120588

119887(120572 120574 120593) = 119891 isin A Ω120588119891 isin R

119887(120572 120574 120593) (83)

Note that the function classR120588119887(120572 120574 120593) is a special case of the

function classR119892119887(120572 120574 120593) when

119892 (119911) = 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119911119899 (84)

Now suppose that

119892 (119911) = 119911 +infin

sum119899=2

119892119899119911119899 (119892

119899gt 0) (85)

Since

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 isin R

119892

119887(120572 120574 120593)

lArrrArr (119891 lowast 119892) (119911) = 119911 +infin

sum119899=2

119892119899119886119899119911119899 isin R

119887(120572 120574 120593)

(86)

we can obtain the coefficient estimates for functions inthe class R119892

119887(120572 120574 120593) from the corresponding estimates for

functions in the class R119887(120572 120574 120593) By applying Theorem 5 to

the following Hadamard product (or convolution)

(119891 lowast 119892) (119911) = 119911 + 119892211988621199112 + 119892

311988631199113 + sdot sdot sdot (87)

we get the following theorem after an obvious change of theparameter 120583

Theorem 15 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (88)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (89)

International Journal of Analysis 7

where

1198611gt 0 119861

2ge 0 119892

119899gt 0 (119899 = 3 4 ) (90)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574) 1198923

(1198612

1198611

minus 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574) 1198923

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574) 1198923

(minus1198612

1198611

+ 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(91)

where

1205901=11989222

1198923

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=11989222

1198923

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(92)

Each of these results is sharp

When 119892 corresponds to the Owa-Srivastava operatorgiven in (82) we obtain

1198922=Γ (3) Γ (2 minus 120588)

Γ (3 minus 120588)=

2

2 minus 120588 (93)

1198923=Γ (4) Γ (2 minus 120588)

Γ (4 minus 120588)=

6

(2 minus 120588) (3 minus 120588) (94)

For 1198922and 1198923given by (93) and (94) respectivelyTheorem 15

reduces to the following result

Theorem 16 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (95)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (96)

where

1198611gt 0 119861

2ge 0 (97)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(1198612

1198611

minus 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(98)

where

1205901=2 (3 minus 120588)

3 (2 minus 120588)

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=2 (3 minus 120588)

3 (2 minus 120588)

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(99)

Each of these results is sharp

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] M Fekete and G Szego ldquoEine bemerkung uber ungeradeschlichte funktionenrdquo Journal of the London MathematicalSociety vol 8 pp 85ndash89 1933

[2] H R Abdel-Gawad and D K Thomas ldquoThe Fekete-Szegoproblem for strongly close-to-convex functionsrdquo Proceedings ofthe American Mathematical Society vol 114 no 2 pp 345ndash3491992

[3] H S Al-Amiri ldquoCertain generalizations of prestarlike func-tionsrdquoAustralianMathematical Society A vol 28 no 3 pp 325ndash334 1979

[4] J H Choi Y C Kim and T Sugawa ldquoA general approach to theFekete-Szego problemrdquo Journal of the Mathematical Society ofJapan vol 59 no 3 pp 707ndash727 2007

[5] A Chonweerayoot D K Thomas and W UpakarnitikasetldquoOn the Fekete-Szego theorem for close-to-convex functionsrdquoInstitut Mathematique vol 66 pp 18ndash26 1992

[6] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquo Mathematica Japonica vol 44 no3 pp 507ndash511 1996

[7] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquoMathematica Japonica vol 47 no 1pp 125ndash132 1998

[8] S Kanas and A Lecko ldquoOn the Fekete-Szego problem and thedomain of convexity for a certain class of univalent functionsrdquoFolia ScientiarumUniversitatis Technicae Resoviensis no 10 pp49ndash57 1990

8 International Journal of Analysis

[9] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[10] W Koepf ldquoOn the Fekete-Szego problem for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 101 no 1 pp 89ndash95 1987

[11] R R London ldquoFekete-Szego inequalities for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 117 no 4 pp 947ndash950 1993

[12] 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

[13] D Bansal ldquoFekete-Szego problem for a new class of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2011 Article ID 143095 5 pages 2011

[14] D Bansal ldquoUpper bound of second Hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[15] G Murugusundaramoorthy and N Magesh ldquoCoefficientinequalities for certain classes of analytic functions associatedwith Hankel determinantrdquo Bulletin of Mathematical Analysisand Applications vol 1 no 3 pp 85ndash89 2009

[16] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962

[17] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences (FJMS) vol 15 no 2 pp171ndash180 2004

[18] S Owa ldquoOn the distortion theorems Irdquo Kyungpook Mathemat-ical Journal vol 18 no 1 pp 53ndash59 1978

[19] H M Srivastava and S Owa Univalent Functions FractionalCalculus and Their Applications Halsted Press (Ellis HorwoodLimited Chichester) John Wiley and Sons New York NYUSA 1989

[20] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987

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 Journal of Analysis

From (46) we have

ℎ1=1

119887(1 + 120572) 119886

2 ℎ

2=1

119887(1 + 2120572 + 2120574) 119886

3 (47)

Since 120593(119911) is univalent and ℎ(119911) ≺ 120593(119911) the function

1199011(119911) =

1 + 120593minus1 (ℎ (119911))

1 minus 120593minus1 (ℎ (119911))= 1 + 119888

1119911 + 11988821199112 + 119888

31199113 + sdot sdot sdot

(48)

is analytic and has a positive real part in U Also we have

ℎ (119911) = 120593(1199011(119911) minus 1

1199011(119911) + 1

)

= 1 +11986111198881

2119911 + [

1198611

2(1198882minus11988821

2) +

119861211988821

4] 1199112 + sdot sdot sdot

(49)

Thus by (47) and (49) we get

1198862=

11986111198881119887

2 (1 + 120572)

1198863=

119887

(1 + 2120572 + 2120574)[1198611

2(1198882minus11988821

2) +

119861211988821

4]

(50)

Taking into account (50) we obtain

1198863minus 12058311988622=

119887

(1 + 2120572 + 2120574)

times [1198611

2(1198882minus11988821

2) +

119861211988821

4] minus 120583

11986121119888211198872

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)(1198882minus 12057511988821)

(51)

where

120575 =1

2(1 minus

1198612

1198611

+1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2) (52)

The assertion of Theorem 5 now follows by an application ofLemma 3 On the other hand using (51) for the values of 120590

1le

120583 le 1205903 we have100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816 + (120583 minus 1205901)1003816100381610038161003816119886210038161003816100381610038162

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (120583 minus 1205901)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (120583 minus(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)100381610038161003816100381610038161198882 minus 120575119888

2

1

10038161003816100381610038161003816 + 1205751003816100381610038161003816119888110038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(53)

Similarly for the values of 1205903le 120583 le 120590

2 we get

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 + (1205902 minus 120583)1003816100381610038161003816119886210038161003816100381610038162

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

+ (1205902minus 120583)

1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)

100381610038161003816100381610038161198882 minus 1205751198882

1

10038161003816100381610038161003816

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

11986121119887 (1 + 2120572 + 2120574)

minus 120583)1198612111988721003816100381610038161003816119888110038161003816100381610038162

4(1 + 120572)2

=1198611119887

2 (1 + 2120572 + 2120574)100381610038161003816100381610038161198882 minus 120575119888

2

1

10038161003816100381610038161003816 + (1 minus 120575)1003816100381610038161003816119888110038161003816100381610038162

le1198611119887

(1 + 2120572 + 2120574)

(54)

To show that the bounds asserted byTheorem 5 are sharp wedefine the following functions

119870120593119899

(119911) (119899 = 2 3 ) (55)

with119870120593119899

(0) = 0 = 1198701015840

120593119899

(0) minus 1 (56)

by

1 +1

119887((1 minus 120572 + 2120574)

119870120593119899

(119911)

119911+ (120572 minus 2120574)1198701015840

120593119899

(119911)

+ 12057411991111987010158401015840120593119899

(119911) minus 1) = 120593 (119911119899minus1)

(57)

International Journal of Analysis 5

and the functions 119865120578(119911) and 119866

120578(119911) (0 le 120578 le 1) with

119865120578(0) = 0 = 119865

1015840

120578(0) minus 1 119866

120578(0) = 0 = 119866

1015840

120578(0) minus 1

(58)

by

1 +1

119887((1 minus 120572 + 2120574)

119865120578(119911)

119911+ (120572 minus 2120574) 1198651015840

120578(119911)

+12057411991111986510158401015840120578(119911) minus 1) = 120593(

119911 (119911 + 120578)

1 + 120578119911)

1 +1

119887((1 minus 120572 + 2120574)

119866120578(119911)

119911+ (120572 minus 2120574)1198661015840

120578(119911)

+ 12057411991111986610158401015840120578(119911) minus 1) = 120593(minus

119911 (119911 + 120578)

1 + 120578119911)

(59)

respectively Then clearly the functions 119870120593119899

119865120578 and 119866

120578isin

R119887(120572 120574 120593) We also write

119870120593= 1198701205932

(60)

If 120583 lt 1205901or 120583 gt 120590

2 then the equality inTheorem 5 holds true

if and only if119891 is119870120593or one of its rotationsWhen120590

1lt 120583 lt 120590

2

then the equality holds true if and only if119891 is1198701205933

or one of itsrotations If 120583 = 120590

1 then the equality holds true if and only

if 119891 is 119865120578or one of its rotations If 120583 = 120590

2 then the equality

holds true if and only if 119891 is 119866120578or one of its rotations

By making use of Lemma 4 we immediately obtain thefollowing Fekete-Szego type inequality

Theorem 6 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (61)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (62)

where

1198611gt 0 119861

2ge 0 (63)

If 119891(119911) given by (1) belongs to the function class R119887(120572 120574 120593)

then for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

(1 + 2120572 + 2120574)max1

100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(64)

The result is sharp

Remark 7 The coefficient bounds for |1198862| and |119886

3| are special

cases of those asserted byTheorem 5Taking 120572 = 2120574 + 1 in Theorem 6 we have the following

corollary

Corollary 8 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (65)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (66)

where

1198611gt 0 119861

2ge 0 (67)

If 119891(119911) given by (1) belongs to the function class R119887120574(120593) then

for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

3 (1 + 2120574)max1

1003816100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus31205831198611119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(68)

The result is sharp

If we set

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (69)

in Theorem 6 then we have

1198611= 119860 minus 119861 119861

2= minus119861 (119860 minus 119861) (70)

So we get the following corollary

Corollary 9 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (71)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (72)

If 119891(119911) given by (1) belongs to the function classR119887(120572 120574 119860 119861)

then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

(1 + 2120572 + 2120574)

timesmax1100381610038161003816100381610038161003816100381610038161003816119861 +

120583 (119860 minus 119861) 119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(73)

The result is sharp

Putting 120572 = 2120574+1 in Corollary 9 we obtain the followingcorollary

Corollary 10 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (74)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (75)

6 International Journal of Analysis

If119891(119911) given by (1) belongs to the function classR119887120574(119860 119861) then

for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

3 (1 + 2120574)

timesmax11003816100381610038161003816100381610038161003816100381610038161003816119861 +

3120583 (119860 minus 119861) 119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(76)

The result is sharp

Also putting 120574 = 0 119887 = 1 119860 = 1 and 119861 = minus1 inCorollary 9 we obtain the following corollary

Corollary 11 Let 120572 gt 0 If 119891(119911) given by (1) belongs to thefunction classR(120572) then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le2

(1 + 2120572)max1

1003816100381610038161003816100381610038161003816100381610038161 minus

2120583 (1 + 2120572)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816 (77)

3 Applications to Analytic FunctionsDefined by Using Fractional CalculusOperators and Convolution

The subject of fractional calculus (ie calculus of integralsand derivatives of any arbitrary real or complex order) hasgained considerable popularity and importance during thepast three decades or so For the applications of the resultsgiven in the preceding sections we first introduce the classR120588

119887(120572 120574 120593) which is defined by means of the Hadamard

product (or convolution) and a certain operator of fractionalcalculus known as the Owa-Srivastava operator (see eg[18 19])

Definition 12 The fractional integral of order 120588 is defined fora function 119891(119911) by

119863minus120588119911119891 (119911) =

1

Γ (120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)1minus120588

119889120577 (120588 gt 0) (78)

where the function 119891(119911) is analytic in a simply connecteddomain of the complex 119911-plane containing the origin and themultiplicity of (119911minus120577)120588minus1 is removed by requiring log(119911minus120577) tobe real when 119911 minus 120577 gt 0

Definition 13 The fractional derivative of order 120588 is definedfor a function 119891(119911) by

119863120588119911119891 (119911) =

1

Γ (1 minus 120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)120588119889120577 (0 le 120588 lt 1)

(79)

where 119891(119911) is constrained and the multiplicity of (119911 minus 120577)minus120588 isremoved as in Definition 12

Definition 14 Under the hypotheses of Definition 13 thefractional derivative of order 119899 + 120588 is defined for a function119891(119911) by

119863119899+120588119911

119891 (119911) =119889119899

119889119911119899(119863120588119911119891 (119911))

(0 le 120588 lt 1 119899 isin N0= N cup 0)

(80)

UsingDefinitions 12 13 and 14 fractional derivatives andfractional integrals Owa and Srivastava [20] introduced theoperatorΩ120588 A rarr A defined by

(Ω120588119891) (119911) = Γ (2 minus 120588) 119911120588119863120588119911119891 (119911) 120588 = 2 3 4 (81)

= 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119886119899119911119899 (82)

This operator is known as the Owa-Srivastava operator Interms of the Owa-Srivastava operator Ω120588 defined by (81) wenow introduce the function classR120588

119887(120572 120574 120593) in the following

way

R120588

119887(120572 120574 120593) = 119891 isin A Ω120588119891 isin R

119887(120572 120574 120593) (83)

Note that the function classR120588119887(120572 120574 120593) is a special case of the

function classR119892119887(120572 120574 120593) when

119892 (119911) = 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119911119899 (84)

Now suppose that

119892 (119911) = 119911 +infin

sum119899=2

119892119899119911119899 (119892

119899gt 0) (85)

Since

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 isin R

119892

119887(120572 120574 120593)

lArrrArr (119891 lowast 119892) (119911) = 119911 +infin

sum119899=2

119892119899119886119899119911119899 isin R

119887(120572 120574 120593)

(86)

we can obtain the coefficient estimates for functions inthe class R119892

119887(120572 120574 120593) from the corresponding estimates for

functions in the class R119887(120572 120574 120593) By applying Theorem 5 to

the following Hadamard product (or convolution)

(119891 lowast 119892) (119911) = 119911 + 119892211988621199112 + 119892

311988631199113 + sdot sdot sdot (87)

we get the following theorem after an obvious change of theparameter 120583

Theorem 15 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (88)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (89)

International Journal of Analysis 7

where

1198611gt 0 119861

2ge 0 119892

119899gt 0 (119899 = 3 4 ) (90)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574) 1198923

(1198612

1198611

minus 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574) 1198923

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574) 1198923

(minus1198612

1198611

+ 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(91)

where

1205901=11989222

1198923

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=11989222

1198923

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(92)

Each of these results is sharp

When 119892 corresponds to the Owa-Srivastava operatorgiven in (82) we obtain

1198922=Γ (3) Γ (2 minus 120588)

Γ (3 minus 120588)=

2

2 minus 120588 (93)

1198923=Γ (4) Γ (2 minus 120588)

Γ (4 minus 120588)=

6

(2 minus 120588) (3 minus 120588) (94)

For 1198922and 1198923given by (93) and (94) respectivelyTheorem 15

reduces to the following result

Theorem 16 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (95)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (96)

where

1198611gt 0 119861

2ge 0 (97)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(1198612

1198611

minus 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(98)

where

1205901=2 (3 minus 120588)

3 (2 minus 120588)

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=2 (3 minus 120588)

3 (2 minus 120588)

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(99)

Each of these results is sharp

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] M Fekete and G Szego ldquoEine bemerkung uber ungeradeschlichte funktionenrdquo Journal of the London MathematicalSociety vol 8 pp 85ndash89 1933

[2] H R Abdel-Gawad and D K Thomas ldquoThe Fekete-Szegoproblem for strongly close-to-convex functionsrdquo Proceedings ofthe American Mathematical Society vol 114 no 2 pp 345ndash3491992

[3] H S Al-Amiri ldquoCertain generalizations of prestarlike func-tionsrdquoAustralianMathematical Society A vol 28 no 3 pp 325ndash334 1979

[4] J H Choi Y C Kim and T Sugawa ldquoA general approach to theFekete-Szego problemrdquo Journal of the Mathematical Society ofJapan vol 59 no 3 pp 707ndash727 2007

[5] A Chonweerayoot D K Thomas and W UpakarnitikasetldquoOn the Fekete-Szego theorem for close-to-convex functionsrdquoInstitut Mathematique vol 66 pp 18ndash26 1992

[6] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquo Mathematica Japonica vol 44 no3 pp 507ndash511 1996

[7] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquoMathematica Japonica vol 47 no 1pp 125ndash132 1998

[8] S Kanas and A Lecko ldquoOn the Fekete-Szego problem and thedomain of convexity for a certain class of univalent functionsrdquoFolia ScientiarumUniversitatis Technicae Resoviensis no 10 pp49ndash57 1990

8 International Journal of Analysis

[9] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[10] W Koepf ldquoOn the Fekete-Szego problem for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 101 no 1 pp 89ndash95 1987

[11] R R London ldquoFekete-Szego inequalities for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 117 no 4 pp 947ndash950 1993

[12] 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

[13] D Bansal ldquoFekete-Szego problem for a new class of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2011 Article ID 143095 5 pages 2011

[14] D Bansal ldquoUpper bound of second Hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[15] G Murugusundaramoorthy and N Magesh ldquoCoefficientinequalities for certain classes of analytic functions associatedwith Hankel determinantrdquo Bulletin of Mathematical Analysisand Applications vol 1 no 3 pp 85ndash89 2009

[16] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962

[17] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences (FJMS) vol 15 no 2 pp171ndash180 2004

[18] S Owa ldquoOn the distortion theorems Irdquo Kyungpook Mathemat-ical Journal vol 18 no 1 pp 53ndash59 1978

[19] H M Srivastava and S Owa Univalent Functions FractionalCalculus and Their Applications Halsted Press (Ellis HorwoodLimited Chichester) John Wiley and Sons New York NYUSA 1989

[20] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987

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 Journal of Analysis 5

and the functions 119865120578(119911) and 119866

120578(119911) (0 le 120578 le 1) with

119865120578(0) = 0 = 119865

1015840

120578(0) minus 1 119866

120578(0) = 0 = 119866

1015840

120578(0) minus 1

(58)

by

1 +1

119887((1 minus 120572 + 2120574)

119865120578(119911)

119911+ (120572 minus 2120574) 1198651015840

120578(119911)

+12057411991111986510158401015840120578(119911) minus 1) = 120593(

119911 (119911 + 120578)

1 + 120578119911)

1 +1

119887((1 minus 120572 + 2120574)

119866120578(119911)

119911+ (120572 minus 2120574)1198661015840

120578(119911)

+ 12057411991111986610158401015840120578(119911) minus 1) = 120593(minus

119911 (119911 + 120578)

1 + 120578119911)

(59)

respectively Then clearly the functions 119870120593119899

119865120578 and 119866

120578isin

R119887(120572 120574 120593) We also write

119870120593= 1198701205932

(60)

If 120583 lt 1205901or 120583 gt 120590

2 then the equality inTheorem 5 holds true

if and only if119891 is119870120593or one of its rotationsWhen120590

1lt 120583 lt 120590

2

then the equality holds true if and only if119891 is1198701205933

or one of itsrotations If 120583 = 120590

1 then the equality holds true if and only

if 119891 is 119865120578or one of its rotations If 120583 = 120590

2 then the equality

holds true if and only if 119891 is 119866120578or one of its rotations

By making use of Lemma 4 we immediately obtain thefollowing Fekete-Szego type inequality

Theorem 6 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (61)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (62)

where

1198611gt 0 119861

2ge 0 (63)

If 119891(119911) given by (1) belongs to the function class R119887(120572 120574 120593)

then for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

(1 + 2120572 + 2120574)max1

100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus1205831198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(64)

The result is sharp

Remark 7 The coefficient bounds for |1198862| and |119886

3| are special

cases of those asserted byTheorem 5Taking 120572 = 2120574 + 1 in Theorem 6 we have the following

corollary

Corollary 8 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (65)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (66)

where

1198611gt 0 119861

2ge 0 (67)

If 119891(119911) given by (1) belongs to the function class R119887120574(120593) then

for any complex number 120583100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le1198611|119887|

3 (1 + 2120574)max1

1003816100381610038161003816100381610038161003816100381610038161003816

1198612

1198611

minus31205831198611119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(68)

The result is sharp

If we set

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (69)

in Theorem 6 then we have

1198611= 119860 minus 119861 119861

2= minus119861 (119860 minus 119861) (70)

So we get the following corollary

Corollary 9 Let

120572 gt 0 0 le 120574 le 1 119887 isin C 0 (71)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (72)

If 119891(119911) given by (1) belongs to the function classR119887(120572 120574 119860 119861)

then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

(1 + 2120572 + 2120574)

timesmax1100381610038161003816100381610038161003816100381610038161003816119861 +

120583 (119860 minus 119861) 119887 (1 + 2120572 + 2120574)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816

(73)

The result is sharp

Putting 120572 = 2120574+1 in Corollary 9 we obtain the followingcorollary

Corollary 10 (see [13]) Let

0 le 120574 le 1 119887 isin C 0 (74)

Also let

120593 (119911) =1 + 119860119911

1 + 119861119911(minus1 le 119861 lt 119860 le 1 119911 isin U) (75)

6 International Journal of Analysis

If119891(119911) given by (1) belongs to the function classR119887120574(119860 119861) then

for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

3 (1 + 2120574)

timesmax11003816100381610038161003816100381610038161003816100381610038161003816119861 +

3120583 (119860 minus 119861) 119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(76)

The result is sharp

Also putting 120574 = 0 119887 = 1 119860 = 1 and 119861 = minus1 inCorollary 9 we obtain the following corollary

Corollary 11 Let 120572 gt 0 If 119891(119911) given by (1) belongs to thefunction classR(120572) then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le2

(1 + 2120572)max1

1003816100381610038161003816100381610038161003816100381610038161 minus

2120583 (1 + 2120572)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816 (77)

3 Applications to Analytic FunctionsDefined by Using Fractional CalculusOperators and Convolution

The subject of fractional calculus (ie calculus of integralsand derivatives of any arbitrary real or complex order) hasgained considerable popularity and importance during thepast three decades or so For the applications of the resultsgiven in the preceding sections we first introduce the classR120588

119887(120572 120574 120593) which is defined by means of the Hadamard

product (or convolution) and a certain operator of fractionalcalculus known as the Owa-Srivastava operator (see eg[18 19])

Definition 12 The fractional integral of order 120588 is defined fora function 119891(119911) by

119863minus120588119911119891 (119911) =

1

Γ (120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)1minus120588

119889120577 (120588 gt 0) (78)

where the function 119891(119911) is analytic in a simply connecteddomain of the complex 119911-plane containing the origin and themultiplicity of (119911minus120577)120588minus1 is removed by requiring log(119911minus120577) tobe real when 119911 minus 120577 gt 0

Definition 13 The fractional derivative of order 120588 is definedfor a function 119891(119911) by

119863120588119911119891 (119911) =

1

Γ (1 minus 120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)120588119889120577 (0 le 120588 lt 1)

(79)

where 119891(119911) is constrained and the multiplicity of (119911 minus 120577)minus120588 isremoved as in Definition 12

Definition 14 Under the hypotheses of Definition 13 thefractional derivative of order 119899 + 120588 is defined for a function119891(119911) by

119863119899+120588119911

119891 (119911) =119889119899

119889119911119899(119863120588119911119891 (119911))

(0 le 120588 lt 1 119899 isin N0= N cup 0)

(80)

UsingDefinitions 12 13 and 14 fractional derivatives andfractional integrals Owa and Srivastava [20] introduced theoperatorΩ120588 A rarr A defined by

(Ω120588119891) (119911) = Γ (2 minus 120588) 119911120588119863120588119911119891 (119911) 120588 = 2 3 4 (81)

= 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119886119899119911119899 (82)

This operator is known as the Owa-Srivastava operator Interms of the Owa-Srivastava operator Ω120588 defined by (81) wenow introduce the function classR120588

119887(120572 120574 120593) in the following

way

R120588

119887(120572 120574 120593) = 119891 isin A Ω120588119891 isin R

119887(120572 120574 120593) (83)

Note that the function classR120588119887(120572 120574 120593) is a special case of the

function classR119892119887(120572 120574 120593) when

119892 (119911) = 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119911119899 (84)

Now suppose that

119892 (119911) = 119911 +infin

sum119899=2

119892119899119911119899 (119892

119899gt 0) (85)

Since

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 isin R

119892

119887(120572 120574 120593)

lArrrArr (119891 lowast 119892) (119911) = 119911 +infin

sum119899=2

119892119899119886119899119911119899 isin R

119887(120572 120574 120593)

(86)

we can obtain the coefficient estimates for functions inthe class R119892

119887(120572 120574 120593) from the corresponding estimates for

functions in the class R119887(120572 120574 120593) By applying Theorem 5 to

the following Hadamard product (or convolution)

(119891 lowast 119892) (119911) = 119911 + 119892211988621199112 + 119892

311988631199113 + sdot sdot sdot (87)

we get the following theorem after an obvious change of theparameter 120583

Theorem 15 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (88)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (89)

International Journal of Analysis 7

where

1198611gt 0 119861

2ge 0 119892

119899gt 0 (119899 = 3 4 ) (90)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574) 1198923

(1198612

1198611

minus 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574) 1198923

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574) 1198923

(minus1198612

1198611

+ 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(91)

where

1205901=11989222

1198923

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=11989222

1198923

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(92)

Each of these results is sharp

When 119892 corresponds to the Owa-Srivastava operatorgiven in (82) we obtain

1198922=Γ (3) Γ (2 minus 120588)

Γ (3 minus 120588)=

2

2 minus 120588 (93)

1198923=Γ (4) Γ (2 minus 120588)

Γ (4 minus 120588)=

6

(2 minus 120588) (3 minus 120588) (94)

For 1198922and 1198923given by (93) and (94) respectivelyTheorem 15

reduces to the following result

Theorem 16 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (95)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (96)

where

1198611gt 0 119861

2ge 0 (97)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(1198612

1198611

minus 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(98)

where

1205901=2 (3 minus 120588)

3 (2 minus 120588)

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=2 (3 minus 120588)

3 (2 minus 120588)

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(99)

Each of these results is sharp

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] M Fekete and G Szego ldquoEine bemerkung uber ungeradeschlichte funktionenrdquo Journal of the London MathematicalSociety vol 8 pp 85ndash89 1933

[2] H R Abdel-Gawad and D K Thomas ldquoThe Fekete-Szegoproblem for strongly close-to-convex functionsrdquo Proceedings ofthe American Mathematical Society vol 114 no 2 pp 345ndash3491992

[3] H S Al-Amiri ldquoCertain generalizations of prestarlike func-tionsrdquoAustralianMathematical Society A vol 28 no 3 pp 325ndash334 1979

[4] J H Choi Y C Kim and T Sugawa ldquoA general approach to theFekete-Szego problemrdquo Journal of the Mathematical Society ofJapan vol 59 no 3 pp 707ndash727 2007

[5] A Chonweerayoot D K Thomas and W UpakarnitikasetldquoOn the Fekete-Szego theorem for close-to-convex functionsrdquoInstitut Mathematique vol 66 pp 18ndash26 1992

[6] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquo Mathematica Japonica vol 44 no3 pp 507ndash511 1996

[7] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquoMathematica Japonica vol 47 no 1pp 125ndash132 1998

[8] S Kanas and A Lecko ldquoOn the Fekete-Szego problem and thedomain of convexity for a certain class of univalent functionsrdquoFolia ScientiarumUniversitatis Technicae Resoviensis no 10 pp49ndash57 1990

8 International Journal of Analysis

[9] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[10] W Koepf ldquoOn the Fekete-Szego problem for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 101 no 1 pp 89ndash95 1987

[11] R R London ldquoFekete-Szego inequalities for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 117 no 4 pp 947ndash950 1993

[12] 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

[13] D Bansal ldquoFekete-Szego problem for a new class of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2011 Article ID 143095 5 pages 2011

[14] D Bansal ldquoUpper bound of second Hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[15] G Murugusundaramoorthy and N Magesh ldquoCoefficientinequalities for certain classes of analytic functions associatedwith Hankel determinantrdquo Bulletin of Mathematical Analysisand Applications vol 1 no 3 pp 85ndash89 2009

[16] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962

[17] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences (FJMS) vol 15 no 2 pp171ndash180 2004

[18] S Owa ldquoOn the distortion theorems Irdquo Kyungpook Mathemat-ical Journal vol 18 no 1 pp 53ndash59 1978

[19] H M Srivastava and S Owa Univalent Functions FractionalCalculus and Their Applications Halsted Press (Ellis HorwoodLimited Chichester) John Wiley and Sons New York NYUSA 1989

[20] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987

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 Journal of Analysis

If119891(119911) given by (1) belongs to the function classR119887120574(119860 119861) then

for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le(119860 minus 119861) |119887|

3 (1 + 2120574)

timesmax11003816100381610038161003816100381610038161003816100381610038161003816119861 +

3120583 (119860 minus 119861) 119887 (1 + 2120574)

4(1 + 120574)2

1003816100381610038161003816100381610038161003816100381610038161003816

(76)

The result is sharp

Also putting 120574 = 0 119887 = 1 119860 = 1 and 119861 = minus1 inCorollary 9 we obtain the following corollary

Corollary 11 Let 120572 gt 0 If 119891(119911) given by (1) belongs to thefunction classR(120572) then for any complex number 120583

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816 le2

(1 + 2120572)max1

1003816100381610038161003816100381610038161003816100381610038161 minus

2120583 (1 + 2120572)

(1 + 120572)2

100381610038161003816100381610038161003816100381610038161003816 (77)

3 Applications to Analytic FunctionsDefined by Using Fractional CalculusOperators and Convolution

The subject of fractional calculus (ie calculus of integralsand derivatives of any arbitrary real or complex order) hasgained considerable popularity and importance during thepast three decades or so For the applications of the resultsgiven in the preceding sections we first introduce the classR120588

119887(120572 120574 120593) which is defined by means of the Hadamard

product (or convolution) and a certain operator of fractionalcalculus known as the Owa-Srivastava operator (see eg[18 19])

Definition 12 The fractional integral of order 120588 is defined fora function 119891(119911) by

119863minus120588119911119891 (119911) =

1

Γ (120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)1minus120588

119889120577 (120588 gt 0) (78)

where the function 119891(119911) is analytic in a simply connecteddomain of the complex 119911-plane containing the origin and themultiplicity of (119911minus120577)120588minus1 is removed by requiring log(119911minus120577) tobe real when 119911 minus 120577 gt 0

Definition 13 The fractional derivative of order 120588 is definedfor a function 119891(119911) by

119863120588119911119891 (119911) =

1

Γ (1 minus 120588)

119889

119889119911int119911

0

119891 (120577)

(119911 minus 120577)120588119889120577 (0 le 120588 lt 1)

(79)

where 119891(119911) is constrained and the multiplicity of (119911 minus 120577)minus120588 isremoved as in Definition 12

Definition 14 Under the hypotheses of Definition 13 thefractional derivative of order 119899 + 120588 is defined for a function119891(119911) by

119863119899+120588119911

119891 (119911) =119889119899

119889119911119899(119863120588119911119891 (119911))

(0 le 120588 lt 1 119899 isin N0= N cup 0)

(80)

UsingDefinitions 12 13 and 14 fractional derivatives andfractional integrals Owa and Srivastava [20] introduced theoperatorΩ120588 A rarr A defined by

(Ω120588119891) (119911) = Γ (2 minus 120588) 119911120588119863120588119911119891 (119911) 120588 = 2 3 4 (81)

= 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119886119899119911119899 (82)

This operator is known as the Owa-Srivastava operator Interms of the Owa-Srivastava operator Ω120588 defined by (81) wenow introduce the function classR120588

119887(120572 120574 120593) in the following

way

R120588

119887(120572 120574 120593) = 119891 isin A Ω120588119891 isin R

119887(120572 120574 120593) (83)

Note that the function classR120588119887(120572 120574 120593) is a special case of the

function classR119892119887(120572 120574 120593) when

119892 (119911) = 119911 +infin

sum119899=2

Γ (119899 + 1) Γ (2 minus 120588)

Γ (119899 minus 120588 + 1)119911119899 (84)

Now suppose that

119892 (119911) = 119911 +infin

sum119899=2

119892119899119911119899 (119892

119899gt 0) (85)

Since

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 isin R

119892

119887(120572 120574 120593)

lArrrArr (119891 lowast 119892) (119911) = 119911 +infin

sum119899=2

119892119899119886119899119911119899 isin R

119887(120572 120574 120593)

(86)

we can obtain the coefficient estimates for functions inthe class R119892

119887(120572 120574 120593) from the corresponding estimates for

functions in the class R119887(120572 120574 120593) By applying Theorem 5 to

the following Hadamard product (or convolution)

(119891 lowast 119892) (119911) = 119911 + 119892211988621199112 + 119892

311988631199113 + sdot sdot sdot (87)

we get the following theorem after an obvious change of theparameter 120583

Theorem 15 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (88)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (89)

International Journal of Analysis 7

where

1198611gt 0 119861

2ge 0 119892

119899gt 0 (119899 = 3 4 ) (90)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574) 1198923

(1198612

1198611

minus 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574) 1198923

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574) 1198923

(minus1198612

1198611

+ 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(91)

where

1205901=11989222

1198923

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=11989222

1198923

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(92)

Each of these results is sharp

When 119892 corresponds to the Owa-Srivastava operatorgiven in (82) we obtain

1198922=Γ (3) Γ (2 minus 120588)

Γ (3 minus 120588)=

2

2 minus 120588 (93)

1198923=Γ (4) Γ (2 minus 120588)

Γ (4 minus 120588)=

6

(2 minus 120588) (3 minus 120588) (94)

For 1198922and 1198923given by (93) and (94) respectivelyTheorem 15

reduces to the following result

Theorem 16 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (95)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (96)

where

1198611gt 0 119861

2ge 0 (97)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(1198612

1198611

minus 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(98)

where

1205901=2 (3 minus 120588)

3 (2 minus 120588)

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=2 (3 minus 120588)

3 (2 minus 120588)

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(99)

Each of these results is sharp

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] M Fekete and G Szego ldquoEine bemerkung uber ungeradeschlichte funktionenrdquo Journal of the London MathematicalSociety vol 8 pp 85ndash89 1933

[2] H R Abdel-Gawad and D K Thomas ldquoThe Fekete-Szegoproblem for strongly close-to-convex functionsrdquo Proceedings ofthe American Mathematical Society vol 114 no 2 pp 345ndash3491992

[3] H S Al-Amiri ldquoCertain generalizations of prestarlike func-tionsrdquoAustralianMathematical Society A vol 28 no 3 pp 325ndash334 1979

[4] J H Choi Y C Kim and T Sugawa ldquoA general approach to theFekete-Szego problemrdquo Journal of the Mathematical Society ofJapan vol 59 no 3 pp 707ndash727 2007

[5] A Chonweerayoot D K Thomas and W UpakarnitikasetldquoOn the Fekete-Szego theorem for close-to-convex functionsrdquoInstitut Mathematique vol 66 pp 18ndash26 1992

[6] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquo Mathematica Japonica vol 44 no3 pp 507ndash511 1996

[7] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquoMathematica Japonica vol 47 no 1pp 125ndash132 1998

[8] S Kanas and A Lecko ldquoOn the Fekete-Szego problem and thedomain of convexity for a certain class of univalent functionsrdquoFolia ScientiarumUniversitatis Technicae Resoviensis no 10 pp49ndash57 1990

8 International Journal of Analysis

[9] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[10] W Koepf ldquoOn the Fekete-Szego problem for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 101 no 1 pp 89ndash95 1987

[11] R R London ldquoFekete-Szego inequalities for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 117 no 4 pp 947ndash950 1993

[12] 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

[13] D Bansal ldquoFekete-Szego problem for a new class of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2011 Article ID 143095 5 pages 2011

[14] D Bansal ldquoUpper bound of second Hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[15] G Murugusundaramoorthy and N Magesh ldquoCoefficientinequalities for certain classes of analytic functions associatedwith Hankel determinantrdquo Bulletin of Mathematical Analysisand Applications vol 1 no 3 pp 85ndash89 2009

[16] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962

[17] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences (FJMS) vol 15 no 2 pp171ndash180 2004

[18] S Owa ldquoOn the distortion theorems Irdquo Kyungpook Mathemat-ical Journal vol 18 no 1 pp 53ndash59 1978

[19] H M Srivastava and S Owa Univalent Functions FractionalCalculus and Their Applications Halsted Press (Ellis HorwoodLimited Chichester) John Wiley and Sons New York NYUSA 1989

[20] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987

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 Journal of Analysis 7

where

1198611gt 0 119861

2ge 0 119892

119899gt 0 (119899 = 3 4 ) (90)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then

100381610038161003816100381610038161198863 minus 1205831198862

2

10038161003816100381610038161003816

le

1198611119887

(1 + 2120572 + 2120574) 1198923

(1198612

1198611

minus 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

1198611119887

(1 + 2120572 + 2120574) 1198923

1205901le 120583 le 120590

2

1198611119887

(1 + 2120572 + 2120574) 1198923

(minus1198612

1198611

+ 1205831198923

11989222

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(91)

where

1205901=11989222

1198923

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=11989222

1198923

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(92)

Each of these results is sharp

When 119892 corresponds to the Owa-Srivastava operatorgiven in (82) we obtain

1198922=Γ (3) Γ (2 minus 120588)

Γ (3 minus 120588)=

2

2 minus 120588 (93)

1198923=Γ (4) Γ (2 minus 120588)

Γ (4 minus 120588)=

6

(2 minus 120588) (3 minus 120588) (94)

For 1198922and 1198923given by (93) and (94) respectivelyTheorem 15

reduces to the following result

Theorem 16 Let

120572 gt 0 0 le 120574 le 1 119887 gt 0 (95)

Also let

120593 (119911) = 1 + 1198611119911 + 11986121199112 + 119861

31199113 + sdot sdot sdot (96)

where

1198611gt 0 119861

2ge 0 (97)

If 119891(119911) given by (1) belongs to the function class R119892119887(120572 120574 120593)

then100381610038161003816100381610038161198863 minus 120583119886

2

2

10038161003816100381610038161003816

le

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(1198612

1198611

minus 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 le 1205901

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)

1205901le 120583 le 120590

2

(2 minus 120588) (3 minus 120588) 1198611119887

6 (1 + 2120572 + 2120574)(minus

1198612

1198611

+ 1205833 (2 minus 120588)

2 (3 minus 120588)

1198611119887 (1 + 2120572 + 2120574)

(1 + 120572)2)

120583 ge 1205902

(98)

where

1205901=2 (3 minus 120588)

3 (2 minus 120588)

(1198612minus 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

1205902=2 (3 minus 120588)

3 (2 minus 120588)

(1198612+ 1198611) (1 + 120572)2

11986121119887 (1 + 2120572 + 2120574)

(99)

Each of these results is sharp

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] M Fekete and G Szego ldquoEine bemerkung uber ungeradeschlichte funktionenrdquo Journal of the London MathematicalSociety vol 8 pp 85ndash89 1933

[2] H R Abdel-Gawad and D K Thomas ldquoThe Fekete-Szegoproblem for strongly close-to-convex functionsrdquo Proceedings ofthe American Mathematical Society vol 114 no 2 pp 345ndash3491992

[3] H S Al-Amiri ldquoCertain generalizations of prestarlike func-tionsrdquoAustralianMathematical Society A vol 28 no 3 pp 325ndash334 1979

[4] J H Choi Y C Kim and T Sugawa ldquoA general approach to theFekete-Szego problemrdquo Journal of the Mathematical Society ofJapan vol 59 no 3 pp 707ndash727 2007

[5] A Chonweerayoot D K Thomas and W UpakarnitikasetldquoOn the Fekete-Szego theorem for close-to-convex functionsrdquoInstitut Mathematique vol 66 pp 18ndash26 1992

[6] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquo Mathematica Japonica vol 44 no3 pp 507ndash511 1996

[7] M Darus and D KThomas ldquoOn the Fekete-Szego theorem forclose-to-convex functionsrdquoMathematica Japonica vol 47 no 1pp 125ndash132 1998

[8] S Kanas and A Lecko ldquoOn the Fekete-Szego problem and thedomain of convexity for a certain class of univalent functionsrdquoFolia ScientiarumUniversitatis Technicae Resoviensis no 10 pp49ndash57 1990

8 International Journal of Analysis

[9] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[10] W Koepf ldquoOn the Fekete-Szego problem for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 101 no 1 pp 89ndash95 1987

[11] R R London ldquoFekete-Szego inequalities for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 117 no 4 pp 947ndash950 1993

[12] 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

[13] D Bansal ldquoFekete-Szego problem for a new class of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2011 Article ID 143095 5 pages 2011

[14] D Bansal ldquoUpper bound of second Hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[15] G Murugusundaramoorthy and N Magesh ldquoCoefficientinequalities for certain classes of analytic functions associatedwith Hankel determinantrdquo Bulletin of Mathematical Analysisand Applications vol 1 no 3 pp 85ndash89 2009

[16] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962

[17] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences (FJMS) vol 15 no 2 pp171ndash180 2004

[18] S Owa ldquoOn the distortion theorems Irdquo Kyungpook Mathemat-ical Journal vol 18 no 1 pp 53ndash59 1978

[19] H M Srivastava and S Owa Univalent Functions FractionalCalculus and Their Applications Halsted Press (Ellis HorwoodLimited Chichester) John Wiley and Sons New York NYUSA 1989

[20] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987

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

8 International Journal of Analysis

[9] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[10] W Koepf ldquoOn the Fekete-Szego problem for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 101 no 1 pp 89ndash95 1987

[11] R R London ldquoFekete-Szego inequalities for close-to-convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 117 no 4 pp 947ndash950 1993

[12] 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

[13] D Bansal ldquoFekete-Szego problem for a new class of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 2011 Article ID 143095 5 pages 2011

[14] D Bansal ldquoUpper bound of second Hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[15] G Murugusundaramoorthy and N Magesh ldquoCoefficientinequalities for certain classes of analytic functions associatedwith Hankel determinantrdquo Bulletin of Mathematical Analysisand Applications vol 1 no 3 pp 85ndash89 2009

[16] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962

[17] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences (FJMS) vol 15 no 2 pp171ndash180 2004

[18] S Owa ldquoOn the distortion theorems Irdquo Kyungpook Mathemat-ical Journal vol 18 no 1 pp 53ndash59 1978

[19] H M Srivastava and S Owa Univalent Functions FractionalCalculus and Their Applications Halsted Press (Ellis HorwoodLimited Chichester) John Wiley and Sons New York NYUSA 1989

[20] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987

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