149–156 DOI: 10.18514/MMN.2018.2118 COEFFICIENT ESTIMATES FOR A CLASS OF ANALYTIC BI-UNIVALENT FUNCTIONS RELATED TO PSEUDO-STARLIKE FUNCTIONS SERAP BULUT Received 13 October, 2016 Abstract

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 149–156 DOI: 10.18514/MMN.2018.2118

COEFFICIENT ESTIMATES FOR A CLASS OF ANALYTIC BI-UNIVALENT FUNCTIONS RELATED TO PSEUDO-STARLIKE

FUNCTIONS

SERAP BULUT Received 13 October, 2016

Abstract. In this paper we introduce and investigate an interesting subclassLB^h;p_˙ ./of ana- lytic and bi-univalent functions in the open unit diskU. For functions belonging to the class LB^h;p_˙ ./, we obtain estimates on the first two Taylor-Maclaurin coefficientsa2anda3. The results presented in this paper would generalize and improve some recent work of Joshi et al. [5].

2010Mathematics Subject Classification: 30C45

Keywords: analytic functions, univalent functions, bi-univalent functions, coefficient estimates, pseudo-starlike functions

1. INTRODUCTION

LetAdenote the class of all functions of the form f .´/D´C

1

X

kD2

ak´^k (1.1)

which are analytic in the open unit disk UD f´W´2C and j´j< 1g: We also denote byS the class of all functions in the normalized analytic function class A which are univalent inU.

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-quarter theorem [4] ensures that the image ofUunder every univalent functionf 2S con- tains a disk of radius1=4:Thus every functionf 2Ahas an inversef ¹;which is defined by

f ¹.f .´//D´ .´2U/

and

f f ¹.w/

Dw

jwj< r0.f /Ir0.f / 1 4

:

c 2018 Miskolc University Press

In fact, the inverse functionf ¹is given by f ¹.w/Dw a2w²C 2a²₂ a3

w³ 5a₂³ 5a2a3Ca4

w⁴C : (1.2) A functionf 2Ais said to be bi-univalent inUif bothf andf ¹are univalent in U, in the sense thatf ¹has a univalent analytic continuation toU:Let˙ denote the class of bi-univalent functions inUgiven by.1:1/ :For a brief history and interesting examples of functions in the class˙; see [6] (see also [2]). In fact, the aforecited work of Srivastavaet al.[6] essentially revived the investigation of various subclasses of the bi-univalent function class˙ in recent years; it was followed by such works as those by Xu et al. [7,8].

Recently, Babalola [1] defined the class L.ˇ/of-pseudo-starlike functions of orderˇas follows:

Suppose0ˇ < 1and1is real. A functionf 2Agiven by.1:1/belongs to the classL.ˇ/of-pseudo-starlike functions of orderˇ in the unit diskUif and only if

< ´ .f⁰.´//

f .´/

!

> ˇ .´2U/ :

Babalola [1] proved that all pseudo-starlike functions are Bazileviˇc of type1 1=, orderˇ¹⁼and univalent inU:

Motivated by this definition, Joshiet al. [5] introduced the following two sub- classes of the bi-univalent function class˙ and obtained non-sharp estimates on the first two Taylor-Maclaurin coefficientsa2 anda3 of functions in each of these sub- classes.

Definition 1 ([5]). A function f .´/ given by .1:1/ is said to be in the class LB_˙.˛/if the following conditions are satisfied:

f 2˙ and ˇ ˇ ˇ ˇ ˇ

arg ´ .f⁰.´//

f .´/

!ˇ ˇ ˇ ˇ ˇ

<˛

2 .´2U/ (1.3)

and ˇ

ˇ ˇ ˇ ˇ

arg w .g⁰.w//

g.w/

!ˇ ˇ ˇ ˇ ˇ

< ˛

2 .w2U/ ; (1.4)

where0 < ˛1; 1and the functiongDf ¹is given by.1:2/ :

We callLB_˙.˛/the class of strongly-bi-pseudo-starlike functions of order˛:

Also forD1, we getLB¹_˙.˛/DS_˙Œ˛the class of strongly bi-starlike functions of order˛;introduced and studied by Brannan and Taha [2].

Theorem 1 ([5]). Let f .´/ given by .1:1/ be in the classLB_˙.˛/ .0 < ˛ 1; 1/. Then

ja2j 2˛

p.2 1/ .2 1C˛/ (1.5)

and

ja3j 4˛²

.2 1/²C 2˛

3 1: (1.6)

Definition 2 ([5]). A function f .´/ given by .1:1/ is said to be in the class LB˙.; ˇ/if the following conditions are satisfied:

f 2˙ and < ´ .f⁰.´//

f .´/

!

> ˇ .´2U/ (1.7) and

< w .g⁰.w//

g.w/

!

> ˇ .w2U/ ; (1.8)

where0ˇ < 1; 1and the functiongDf ¹is defined by.1:2/ :

We callLB˙.; ˇ/the class of-bi-pseudo-starlike functions of orderˇ:Also forD1, we getLB˙.1; ˇ/DS_˙.ˇ/the class of bi-starlike functions of orderˇ;

introduced and studied by Brannan and Taha [2].

Theorem 2([5]). Letf .´/given by.1:1/be in the classLB˙.; ˇ/ .0ˇ <

1; 1/. Then

ja2j s

2 .1 ˇ/

.2 1/ (1.9)

and

ja3j 4 .1 ˇ/²

.2 1/² C2 .1 ˇ/

3 1 : (1.10)

Here, in our present paper, inspiring by some of the aforecited works (especially [5]), we introduce the following subclass of the analytic function classA, analog- ously to the definition given by Xu et al. [7].

Definition 3. Let the functionsh; pWU!Cbe so constrained that minf<.h .´// ;<.p .´//g> 0 .´2U/ and h .0/Dp .0/D1:

Also let the functionf defined by.1:1/be in the analytic function classA. We say thatf 2LB^h;p_˙ ./ .1/if the following conditions are satisfied:

f 2˙ and ´ .f⁰.´//

f .´/ 2h .U/ .´2U/ (1.11) and

w .g⁰.w//

g.w/ 2p .U/ .w2U/ ; (1.12)

where the functiongDf ¹is defined by.1:2/.

SettingD1in Definition3, we get the classLB^h;p_˙ .1/DB_˙^h;pintroduced and studied by Bulut [3].

Remark1. There are many choices of the functionshandpwhich would provide interesting subclasses of the analytic function classA. For example, if we let

h .´/D

1C´ 1 ´

˛

and p .´/D

1 ´ 1C´

˛

.0 < ˛1; ´2U/

or

h .´/D1C.1 2ˇ/ ´

1 ´ and p .´/D 1 .1 2ˇ/ ´

1C´ .0ˇ < 1; ´2U/ ; it is easy to verify that the functionsh.´/andp.´/satisfy the hypotheses of Defini- tion3:Iff 2LB^h;p_˙ ./ ;then

f 2˙ and ˇ ˇ ˇ ˇ ˇ

arg ´ .f⁰.´//

f .´/

!ˇ ˇ ˇ ˇ ˇ

< ˛

2 .0 < ˛1; 1; ´2U/

and

ˇ ˇ ˇ ˇ ˇ

arg w .g⁰.w//

g.w/

!ˇ ˇ ˇ ˇ ˇ

< ˛

2 .0 < ˛1; 1; w2U/

or

f 2˙ and < ´ .f⁰.´//

f .´/

!

> ˇ .0ˇ < 1; 1; ´2U/

and

< w .g⁰.w//

g.w/

!

> ˇ .0ˇ < 1; 1; w2U/ ;

where the functiongDf ¹is defined by.1:2/ :This means that f 2LB_˙.˛/ .0 < ˛1; 1/

or

f 2LB˙.; ˇ/ .0ˇ < 1; 1/ :

Motivated and stimulated especially by the work of Joshiet al.[5], we propose to investigate the bi-univalent function classLB^h;p_˙ ./introduced in Definition3here and derive coefficient estimates on the first two Taylor-Maclaurin coefficientsa2and a3 for a functionf 2LB^h;p_˙ ./ given by .1:1/. Our results for the bi-univalent function classLB^h;p_˙ ./would generalize and improve the related work of Joshiet al.[5].

2. ASET OF GENERAL COEFFICIENT ESTIMATES

In this section we state and prove our general results involving the bi-univalent function classLB^h;p_˙ ./given by Definition3.

Theorem 3. Let the functionf .´/given by the Taylor-Maclaurin series expansion .1:1/be in the bi-univalent function classLB^h;p_˙ ./ :Then

ja2j min 8

<

: s

jh⁰.0/j²C jp⁰.0/j² 2 .2 1/² ;

s

jh⁰⁰.0/j C jp⁰⁰.0/j 4 .2 1/

9

=

;

(2.1) and

ja3j minf ; ıg; (2.2)

where

Djh⁰.0/j²C jp⁰.0/j²

2 .2 1/² Cjh⁰⁰.0/j C jp⁰⁰.0/j 4 .3 1/

and

ıD 8 ˆ<

ˆ:

2²C2 1

jh⁰⁰.0/j 2² 4C1

jp⁰⁰.0/j ; 1 < 1C

p2 2

2²C2 1

jh⁰⁰.0/j C 2² 4C1

jp⁰⁰.0/j ; 1C

p2 2

4 .2 1/ .3 1/ :

Proof. First of all, we write the argument inequalities in.1:11/and.1:12/in their equivalent forms as follows:

´ .f⁰.´//

f .´/ Dh .´/ .´2U/ ; and

w .g⁰.w//

g.w/ Dp .w/ .w2U/ ;

respectively, whereh .´/ andp .w/satisfy the conditions of Definition 3. Further- more, the functionsh .´/andp .w/have the following Taylor-Maclaurin series ex- pensions:

h.´/D1Ch1´Ch2´²C and

p.w/D1Cp1wCp2w²C ; respectively. Now, upon equating the coefficients of ^´.f0.´//

f .´/ with those ofh.´/and the coefficients of ^w.g0.w//

g.w/ with those ofp .w/, we get

.2 1/ a2Dh1; (2.3)

2² 4C1

a²₂C.3 1/ a3Dh2; (2.4)

.2 1/ a2Dp1 (2.5) and

2²C2 1

a²₂ .3 1/ a3Dp2: (2.6)

From.2:3/and.2:5/ ;we obtain

h1D p1 (2.7)

and

2 .2 1/²a₂²Dh²₁Cp²₁: (2.8) Also, from.2:4/and.2:6/ ;we find that

2 .2 1/ a²₂Dh2Cp2: (2.9)

Therefore, we find from the equations.2:8/and.2:9/that ja2j²jh⁰.0/j²C jp⁰.0/j²

2 .2 1/² and

ja2j²jh⁰⁰.0/j C jp⁰⁰.0/j 4 .2 1/ ;

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, we subtract .2:6/ from .2:4/. We thus get

2 .3 1/ a3 2 .3 1/ a₂²Dh2 p2: (2.10) Upon substituting the value ofa²₂from.2:8/into.2:10/ ;it follows that

a3D h²₁Cp₁²

2 .2 1/²C h2 p2

2 .3 1/: We thus find that

ja3j jh⁰.0/j²C jp⁰.0/j²

2 .2 1/² Cjh⁰⁰.0/j C jp⁰⁰.0/j 4 .3 1/ :

On the other hand, upon substituting the value ofa₂²from.2:9/into.2:10/ ;it follows that

a3D 2²C2 1

h2C 2²C4 1 p2

2 .2 1/ .3 1/ :

We thus obtain

ja₃j 8 ˆ<

ˆ:

2²C2 1

jh⁰⁰.0/j 2² 4C1

jp⁰⁰.0/j ; 1 < 1C

p2 2

2²C2 1

jh⁰⁰.0/j C 2² 4C1

jp⁰⁰.0/j ; 1C

p2 2

4 .2 1/ .3 1/ :

This evidently completes the proof of Theorem3.

3. COROLLARIES AND CONSEQUENCES

SettingD1in Theorem3, we get following consequence.

Corollary 1 ([3]). Let the functionf .´/ given by the Taylor-Maclaurin series expansion.1:1/be in the bi-univalent function classB_˙^h;p:Then

ja2j min 8

<

: s

jh⁰.0/j²C jp⁰.0/j²

2 ;

rjh⁰⁰.0/j C jp⁰⁰.0/j 4

9

=

; and

ja3j min

(jh⁰.0/j²C jp⁰.0/j²

2 Cjh⁰⁰.0/j C jp⁰⁰.0/j

8 ; 3jh⁰⁰.0/j C jp⁰⁰.0/j 8

) :

If we set h .´/D

1C´ 1 ´

˛

and p .´/D

1 ´ 1C´

˛

.0 < ˛1; ´2U/

in Theorem3, we can readily deduce Corollary2.

Corollary 2. Let the functionf .´/given by the Taylor-Maclaurin series expan- sion.1:1/be in the bi-univalent function classLB_˙.˛/ .0 < ˛1; 1/. Then

ja2j s

2˛² .2 1/

and

ja3j 8 ˆ<

ˆ:

2˛²

.2 1/ ; 1 < 1C

p2 2 2˛²

3 1 1C

p2 2

:

Remark2. Corollary2is an improvement of Theorem1:

If we set

h .´/D 1C.1 2ˇ/ ´

1 ´ and p .´/D1 .1 2ˇ/ ´

1C´ .0ˇ < 1; ´2U/

in Theorem3, we can readily deduce Corollary3.

Corollary 3. Let the functionf .´/given by the Taylor-Maclaurin series expan- sion.1:1/be in the bi-univalent function classLB˙.; ˇ/ .0ˇ < 1; 1/. Then

ja2j 8 ˆ<

ˆ:

q 1 ˇ

.2 1/ 0ˇ²₄^C1

2.1 ˇ / .2 1/

2C1

4 ˇ < 1

and

ja3j 8 ˆ<

ˆ: min

n_{4.1 ˇ /}2

.2 1/² C_{3 1}^{1 ˇ} ; _{.2 1/}^{1 ˇ} o

; 1 < 1C

p2 2 1 ˇ

3 1 1C

p2 2

:

Remark3. Corollary3is an improvement of Theorem2.

REFERENCES

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[3] S. Bulut, “Coefficient estimates for a class of analytic and bi-univalent functions,”Novi Sad J.

Math., vol. 43, no. 2, pp. 59–65, 2013.

[4] P. Duren,Univalent Functions, ser. Grundlehren der Mathematischen Wissenschaften. New York:

Springer, 1983, vol. 259.

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[6] H. Srivastava, A. Mishra, and P. Gochhayat, “Certain subclasses of analytic and bi-univalent func- tions,”Appl. Math. Lett., vol. 23, pp. 1188–1192, 2010, doi:10.1016/j.aml.2010.05.009.

[7] Q.-H. Xu, Y.-C. Gui, and H. Srivastava, “Coefficient estimates for a certain subclass of analytic and bi-univalent functions,” Appl. Math. Lett., vol. 25, pp. 990–994, 2012, doi:

10.1016/j.aml.2011.11.013.

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Author’s address

Serap Bulut

Kocaeli University, Faculty of Aviation and Space Sciences, Arslanbey Campus, 41285 Kartepe- Kocaeli, Turkey

E-mail address:serap.bulut@kocaeli.edu.tr