Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 17 (2016), No. 1, pp. 101–110 DOI: 10.18514/MMN.2016.1162

COEFFICIENT ESTIMATES FOR A NEW SUBCLASS OFANALYTIC AND BI-UNIVALENT FUNCTIONS DEFINED BY

CONVOLUTION

SERAP BULUT

Received 10 March, 2014

Abstract. In this paper, we introduce and investigate a new subclass Hh;p˙

.�I�/ of analytic andbi-univalent functions in the open unit disk U: For functions belonging to this class, we obtainestimates on the first two Taylor-Maclaurin coefficients ja2j and ja3j :

2010 Mathematics Subject Classification: 30C45

Keywords: analytic functions, univalent functions, bi-univalent functions, Taylor-Maclaurin seriesexpansion, coefficient bounds and coefficient estimates, Taylor-Maclaurin coefficients, Hadam-ard product (convolution)

1. INTRODUCTION

Let A denote the class of all functions of the form

f .´/D ´C

1XnD2

an´n (1.1)

which are analytic in the open unit disk

U Df´ W ´ 2C and j´j< 1g :

We also denote by S the class of all functions in the normalized analytic functionclass A which are univalent in U:

Since univalent functions are one-to-one, they are invertible and the inverse func-tions need not be defined on the entire unit disk U: In fact, the Koebe one-quartertheorem [7] ensures that the image of U under every univalent function f 2 S con-tains a disk of radius 1=4: Thus every function f 2A has an inverse f �1; which isdefined by

f �1 .f .´//D ´ .´ 2U/

and

f�f �1 .w/

�D w

�jwj< r0 .f / I r0 .f /�

1

4

�:

c 2016 Miskolc University Press

102 SERAP BULUT

In fact, the inverse function f �1 is given by

f �1 .w/D w�a2w2C�2a22�a3

�w3�

�5a32�5a2a3Ca4

�w4C�� � :

Denote by f �� the Hadamard product (or convolution) of the functions f and�; that is, if f is given by (1.1) and � is given by

�.´/D ´C

1XnD2

bn´n .bn > 0/; (1.2)

then

.f ��/.´/D ´C

1XnD2

anbn´n: (1.3)

For two functions f and �, analytic in U; we say that the function f is subordin-ate to � in U; and write

f .´/��.´/ .´ 2U/ ;

if there exists a Schwarz function !; which is analytic in U with

! .0/D 0 and j! .´/j< 1 .´ 2U/

such thatf .´/D�.! .´// .´ 2U/ :

Indeed, it is known that

f .´/��.´/ .´ 2U/) f .0/D�.0/ and f .U/��.U/ :

Furthermore, if the function� is univalent in U; then we have the following equival-ence

f .´/��.´/ .´ 2U/, f .0/D�.0/ and f .U/��.U/ :

A function f 2A is said to be bi-univalent in U if both f and f �1 are univalentin U: Let˙ denote the class of bi-univalent functions in U given by (1.1). For a briefhistory and interesting examples of functions in the class˙; see [13] (see also [2]). Infact, the aforecited work of Srivastava et al. [13] essentially revived the investigationof various subclasses of the bi-univalent function class ˙ in recent years; it wasfollowed by such works as those by El-Ashwah [8], Frasin and Aouf [9], Aouf et al.[1], and others (see, for example, [3–6, 10–12, 14]).

Definition 1. Let the functions h;p WU!C be so constrained that

minf<.h.´// ;<.p .´//g> 0 .´ 2U/ and h.0/D p .0/D 1:

Also let the function f; defined by (1.1), be in the analytic function class A and let� 2˙: We say that

f 2Hh;p˙ .�I�/ .�� 1/

ON A NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS 103

if the following conditions are satisfied:

f 2˙ and .1��/.f ��/.´/

´C�.f ��/0 .´/ 2 h.U/ .´ 2U/ (1.4)

and

.1��/.f ��/�1 .w/

wC�

�.f ��/�1

�0.w/ 2 p .U/ .w 2U/ ; (1.5)

where the function .f ��/�1 is defined by

.f ��/�1 .w/D w�a2b2w2C�2a22b

22 �a3b3

�w3

��5a32b

32 �5a2b2a3b3Ca4b4

�w4C�� � : (1.6)

Remark 1. If we let

�.´/D´

1�´D ´C

1XnD2

´n;

then the class Hh;p˙ .�I�/ reduces to the class denoted by B

h;p˙ .�/ which is the

subclass of the functions f 2˙ satisfying

.1��/f .´/

´C�f 0 .´/ 2 h.U/

and

.1��/g .w/

wC�g0 .w/ 2 p .U/ ;

where the function g is defined by

g .w/D w�a2w2C�2a22�a3

�w3�

�5a32�5a2a3Ca4

�w4C�� � : (1.7)

This class is introduced and studied by Xu et al. [15]. Also we get the function class

Hh;p˙

�1I

´

1�´

�DH

h;p˙

introduced and studied by Xu et al. [14].

Remark 2. If we let

h.´/D

�1C´

1�´

�˛and p .´/D

�1�´

1C´

�˛.0 < ˛ � 1/ ;

then the class Hh;p˙ .�I�/ reduces to the class denoted by B˙ .�;˛;�/ which is the

subclass of the functions f 2˙ satisfyingˇarg�.1��/

.f ��/.´/

´C�.f ��/0 .´/

�ˇ<˛�

2

and ˇˇarg

(.1��/

.f ��/�1 .w/

wC�

�.f ��/�1

�0.w/

)ˇˇ< ˛�

2;

104 SERAP BULUT

where the function .f ��/�1 is defined by (1.6). This class is introduced and studiedby El-Ashwah [8]. In this class,

(i) if we set

�.´/D´

1�´D ´C

1XnD2

´n;

then the class B˙ .�;˛;�/ reduces to the class denoted by B˙ .˛;�/ introduced andstudied by Frasin and Aouf [9]. Also we have the class

B˙ .˛;1/DH˛˙

introduced and studied by Srivastava et al. [13].(ii) if we set

�.´/D ´C

1XnD2

.a1/n � � ��aq�n

.b1/n � � �.bs/n

1

nŠ´n;

then the class B˙ .�;˛;�/ reduces to the class denoted by T ˙q;s Œa1Ib1;˛;�� intro-

duced and studied Aouf et al. [1].

Remark 3. If we let

h.´/D1C .1�2ˇ/´

1�´and p .´/D

1� .1�2ˇ/´

1C´.0� ˇ < 1/;

then the class Hh;p˙ .�I�/ reduces to the class denoted by B˙ .�;ˇ;�/ which is the

subclass of the functions f 2˙ satisfying

<

�.1��/

.f ��/.´/

´C�.f ��/0 .´/

�> ˇ

and

<

(.1��/

.f ��/�1 .w/

wC�

�.f ��/�1

�0.w/

)> ˇ;

where the function .f ��/�1 is defined by (1.6). This class is introduced and studiedby El-Ashwah [8]. In this class,

(i) if we set

�.´/D´

1�´D ´C

1XnD2

´n;

then the class B˙ .�;ˇ;�/ reduces to the class denoted by B˙ .ˇ;�/ introduced andstudied by Frasin and Aouf [9]. Also we have the class

B˙ .ˇ;1/DH˙ .ˇ/

introduced and studied by Srivastava et al. [13].

ON A NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS 105

(ii) if we set

�.´/D ´C

1XnD2

.a1/n � � ��aq�n

.b1/n � � �.bs/n

1

nŠ´n;

then the class B˙ .�;ˇ;�/ reduces to the class denoted by T ˙q;s Œa1Ib1;ˇ;�� intro-

duced and studied by Aouf et al. [1].

2. A SET OF GENERAL COEFFICIENT ESTIMATES

In this section, we state and prove our general results involving the bi-univalentfunction class H

h;p˙ .�I�/ given by Definition 1.

Theorem 1. Let the function f .´/ given by the Taylor-Maclaurin series expansion(1.1) be in the function class

Hh;p˙ .�I�/ .�� 1/

with

� 2˙ and �.´/D ´C

1XnD2

bn´n .bn > 0/:

Then

ja2j �1

b2min

8<:sjh0 .0/j2Cjp0 .0/j2

2.1C�/2;

sjh00 .0/jC jp00 .0/j

4.1C2�/

9=; (2.1)

and

ja3j �1

b3min

(jh0 .0/j

2Cjp0 .0/j

2

2.1C�/2Cjh00 .0/jC jp00 .0/j

4.1C2�/;jh00 .0/j

2.1C2�/

): (2.2)

Proof. First of all, we write the argument inequalities in (1.4) and (1.5) in theirequivalent forms as follows:

.1��/.f ��/.´/

´C�.f ��/0 .´/D h.´/ .´ 2U/

and

.1��/.f ��/�1 .w/

wC�

�.f ��/�1

�0.w/D p .w/ .w 2U/ ;

respectively, where h.´/ and p .w/ satisfy the conditions of Definition 1. Further-more, the functions h.´/ and p .w/ have the following Taylor-Maclaurin series ex-pansions:

h.´/D 1Ch1´Ch2´2C�� �

andp .w/D 1Cp1wCp2w

2C�� � ;

106 SERAP BULUT

respectively. Now, upon equating the coefficients of

.1��/.f ��/.´/

´C�.f ��/0 .´/

with those of h.´/ and the coefficients of

.1��/.f ��/�1 .w/

wC�

�.f ��/�1

�0.w/

with those of p .w/; we get

.1C�/b2a2 D h1; (2.3)

.1C2�/b3a3 D h2; (2.4)

�.1C�/b2a2 D p1 (2.5)

and� .1C2�/b3a3C2.1C2�/b

22a22 D p2: (2.6)

From (2.3) and (2.5), it follows that

h1 D�p1 (2.7)

and2.1C�/2 b22a

22 D h

21Cp

21 : (2.8)

Also from (2.4) and (2.6), we find that

2.1C2�/b22a22 D h2Cp2: (2.9)

Therefore, we find from the equations (2.8) and (2.9) that

ja2j2�jh0 .0/j

2Cjp0 .0/j

2

2.1C�/2 b22

and

ja2j2�jh00 .0/jC jp00 .0/j

4.1C2�/b22;

respectively. So we get the desired estimate on the coefficient ja2j as asserted in(2.1).

Next, in order to find the bound on the coefficient ja3j ; by subtracting (2.6) from(2.4), we obtain

2.1C2�/b3a3�2.1C2�/b22a22 D h2�p2: (2.10)

Upon substituting the value of a22 from (2.8) into (2.10), it follows that

a3 Dh21Cp

21

2.1C�/2 b3C

h2�p2

2.1C2�/b3:

ON A NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS 107

We thus find that

ja3j �jh0 .0/j

2Cjp0 .0/j

2

2.1C�/2 b3Cjh00 .0/jC jp00 .0/j

4.1C2�/b3:

On the other hand, upon substituting the value of a22 from (2.9) into (2.10), it followsthat

a3 Dh2

.1C2�/b3:

Consequently, we have

ja3j �jh00 .0/j

2.1C2�/b3:

This evidently completes the proof of Theorem 1. �

Remark 4. If we take

�.´/D´

1�´D ´C

1XnD2

´n

in Theorem 1, then we have the estimates obtained by Srivastava et al. [12, Corollary1] which is an improvement of the estimates given by Xu et al. [15, Theorem 3]. Inaddition the above condition on �; if we set �D 1; then we get [14, Theorem 3].

If we let

h.´/D

�1C´

1�´

�˛and p .´/D

�1�´

1C´

�˛.0 < ˛ � 1/

in Theorem 1; then we have Corollary 1 below.

Corollary 1. Let the function f .´/ given by the Taylor-Maclaurin series expan-sion (1.1) be in the function class

B˙ .�;˛;�/ .�� 1; 0 < ˛ � 1/

with

� 2˙ and �.´/D ´C

1XnD2

bn´n .bn > 0/:

Then

ja2j �1

b2

8<:

2˛1C�

; �� 1Cp2q

21C2�

˛ ; 1� � < 1Cp2

and

ja3j �2˛2

.1C2�/b3:

108 SERAP BULUT

Remark 5. It is easy to see, for the coefficient ja2j ; that

2˛

1C��

2˛q.1C�/2C˛

�1C2���2

� �0 < ˛ � 1; �� 1C

p2�

andr2

1C2�˛ �

2˛q.1C�/2C˛

�1C2���2

� �0 < ˛ � 1; 1� � < 1C

p2�:

On the other hand, for the coefficient ja3j ; we have

2˛2

1C2��

2˛

1C2�

�4˛2

.1C�/2C

2˛

1C2�.0 < ˛ � 1; �� 1/ :

Thus, clearly, Corollary 1 is an improvement of the estimates obtained by El-Ashwah[8, Theorem 1].

Remark 6. (i) If we take

�.´/D´

1�´D ´C

1XnD2

´n

in Corollary 1, then we have an improvement of the estimates obtained by Frasin andAouf [9, Theorem 2.2]. In addition the above condition on �; if we set �D 1; thenwe get an improvement of the estimates obtained by Srivastava et al. [13, Theorem1].

(ii) If we take

�.´/D ´C

1XnD2

.a1/n � � ��aq�n

.b1/n � � �.bs/n

1

nŠ´n

in Corollary 1, then we have an improvement of the estimates obtained by Aouf et al.[1, Theorem 4].

If we let

h.´/D1C .1�2ˇ/´

1�´and p .´/D

1� .1�2ˇ/´

1C´.0� ˇ < 1/

in Theorem 1, then we have Corollary 2 below.

Corollary 2. Let the function f .´/ given by the Taylor-Maclaurin series expan-sion (1.1) be in the function class

B˙ .�;ˇ;�/ .�� 1; 0� ˇ < 1/

ON A NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS 109

with

� 2˙ and �.´/D ´C

1XnD2

bn´n .bn > 0/:

Then

ja2j �1

b2min

8<:2.1�ˇ/1C�;

s2.1�ˇ/

1C2�

9=;and

ja3j �2.1�ˇ/

.1C2�/b3:

Remark 7. Corollary 2 is an improvement of the estimates obtained by El-Ashwah[8, Theorem 2].

Remark 8. (i) If we take

�.´/D´

1�´D ´C

1XnD2

´n

in Corollary 2, then we have an improvement of the estimates obtained by Frasin andAouf [9, Theorem 3.2]. In addition the above condition on �; if we set �D 1; thenwe get an improvement of the estimates obtained by Srivastava et al. [13, Theorem2].

(ii) If we take

�.´/D ´C

1XnD2

.a1/n � � ��aq�n

.b1/n � � �.bs/n

1

nŠ´n

in Corollary 2, then we have an improvement of the estimates obtained by Aouf et al.[1, Theorem 8].

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Author’s address

Serap BulutKocaeli University, Faculty of Aviation and Space Sciences, Arslanbey Campus, 41285 Kartepe-

Kocaeli, TURKEYE-mail address: serap.bulut@kocaeli.edu.tr