Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 17 (2016), No. 1, pp. 29–34 DOI: 10.18514/MMN.2016.768

COEFFICIENT INEQUALITY FOR CERTAIN CLASSES OFANALYTIC FUNCTIONS

T. AL-HAWARY, B. A. FRASIN, AND M. DARUS

Received 26 January, 2014

Abstract. Let T .s;�;�;ˇ/ be the class of normalized functions, f defined in the open unit diskU by

Re

��;sC1� f .´/

��;s� g.´/

!> 0 .� 2N; �;s 2N0; ´ 2U/

for some g 2Rˇ .s;�;�;ˇ/. The authors in [15] introduced the operator ��;s� defined by

��;s� f .´/D ´C

1XkD2

.kC��1/Š.��1/Š

�Š.kC��2/Šksak´

k ;

where Rˇ .s;�;�;ˇ/ denotes the class of normalized functions g in the open unit disk U definedby ˇ

ˇarg

��;sC1� g.´/

��;s� g.´/

!ˇˇ< �

2ˇ .0 < ˇ � 1/ :

For f 2 T .s;�;�;ˇ/ and given by f .´/ D ´C a2´2C a3´3C �� � ; a sharp upper bound isobtained for

ˇa3� �a

22

ˇwhen � � 1:

2010 Mathematics Subject Classification: 30C45

Keywords: univalent and analytic functions, starlike and convex functions, Fekete-Szego prob-lem

1. INTRODUCTION AND DEFINITIONS

Let A denote the family of functions of the form:

f .´/D ´C

1XkD2

ak´k; (1.1)

which are analytic in the open unit disk U D f´ W j´j < 1g: Further, let S denote theclass of functions which are univalent in U: A function f .´/ belonging to A is said

The work was fully supported by AP-2013-009.

c 2016 Miskolc University Press

30 T. AL-HAWARY, B. A. FRASIN, AND M. DARUS

to be strongly starlike of order ˇ in U , and denoted by SS�.ˇ/ if it satisfiesˇarg�´f 0.´/

f .´/

�ˇ<�

2ˇ .0 < ˇ � 1; ´ 2U/: (1.2)

If f .´/ 2A satisfiesˇarg�1C

´f 00.´/

f 0.´/

�ˇ<�

2ˇ .0 < ˇ � 1; ´ 2U/; (1.3)

then we say that f .´/ is strongly convex of order ˇ in U , and we denote by SC.ˇ/

the class of all such functions.With the help of the differential operator��;s� ;we say that a function f .´/ belongingto A is said to be in the class R

ˇ.s;�;�;ˇ/ if it satisfiesˇ

ˇarg

��;sC1� f .´/

��;s� f .´/

!ˇˇ< �

2ˇ .� 2N;�;s 2N0 DN[f0g/; (1.4)

for some ˇ.0 < ˇ � 1/ and for all ´ 2 U: Note that Rˇ.0;0;1;ˇ/ D SS�.ˇ/ and

Rˇ.1;0;1;ˇ/D SC.ˇ/.

For the class S of analytic univalent functions, [6] obtained the maximum value ofˇa3� �a

22

ˇwhen � is real. For various functions of S ; the upper bound for

ˇa3� �a

22

ˇis investigated by many different authors (see [1–5, 7, 8, 11–14, 16, 17] and [18]).

In this paper we obtain a sharp upper bounds forˇa3� �a

22

ˇwhen f belongs to

the class of functions defined as follows:

Definition 1. Let ˇ.0 < ˇ � 1/ and letf 2A. Then f 2 T .s;�;�;ˇ/ if and onlyif there exist g 2R

ˇ.s;�;�;ˇ/ such that

Re

��;sC1� f .´/

��;s� g.´/

!> 0 .� 2N;�;s 2N0 DN[f0g; ´ 2U/, (1.5)

where g.´/D ´Cb2´2Cb3´3C�� � :

Note that T .0;0;1;ˇ/DK.ˇ/ the class of close-to-convex functions defined by[3], T .0;0;1;1/DK.1/ is the class of normalized close-to-convex functions definedby [10].

2. MAIN RESULTS

In order to derive our main results, we have to recall here the following lemma.

Lemma 1 ([17]). Let h 2 P i.e. h be analytic in U and be given by h.´/ D1C c1´C c2´

2C c3´3C�� � , and Reh.´/ > 0 for ´ 2 U: Thenˇ

c2�c212

ˇ� 2�

ˇc21ˇ

2: (2.1)

COEFFICIENT INEQUALITY FOR CERTAIN CLASSES... 31

Theorem 1. Let f .´/ 2 T .s;�;�;ˇ/ and be given by (1.1). Then for ˇ � 1 and� � 1 we have the sharp inequalityˇ

a3� �a22

ˇ�ˇ2�.3s��2.�C2/�22s�.�C1/.�C1/

�22s3s.�C2/.�C1/2

C

�.3sC1��2.�C2/�22sC1�.�C1/.�C1/

�.2ˇC1/

22s3sC1.�C2/.�C1/2: (2.2)

Proof. Let f .´/ 2 T .s;�;�;ˇ/: It follows from (1.5) that

��;sC1� f .´/D��;s� g.´/q.´/; (2.3)

for ´ 2U, with q 2P given by q.´/D 1Cq1´Cq2´2Cq3´3C�� � : Equating coef-ficients, we obtain

.�C1/

�2sC1a2 D q1C

.�C1/

�2sb2; (2.4)

and.�C2/.�C1/

�.�C1/3sC1a3 D q2C

.�C1/

�2sb2q1C

.�C2/.�C1/

�.�C1/3sb3: (2.5)

Also, it follows from (1.4) that

��;sC1� g.´/D��;s� g.´/.p.´//ˇ

:

where for ´ 2 U; p 2 P and p.´/ D 1Cp1´Cp2´2Cp3´3C�� � . Thus equatingcoefficients, we obtain

.�C1/

�2sb2 D ˇp1; (2.6)

.�C2/.�C1/

�.�C1/.2/3sb3 D ˇ.p2C

3ˇ�1

2p21/: (2.7)

From (2.4), (2.5), (2.6) and (2.7) we have

a3� �a22 D

�.�C1/

3sC1.�C2/.�C1/.q2�

1

2q21/

C22sC1�.�C1/.�C1/�3sC1��2.�C2/

3sC1�22sC2.�C2/.�C1/2q21

Cˇ�.�C1/

2�3sC1.�C2/.�C1/.p2�

1

2p21/

CŒ22sC1�.�C1/.�C1/�3sC1��2.�C2/�ˇ

3sC1�22sC1.�C2/.�C1/2p1q1

C3�22sˇ2�.�C1/.�C1/�3sC1�ˇ2�2.�C2/

3sC122sC2.�C2/.�C1/2p21 : (2.8)

32 T. AL-HAWARY, B. A. FRASIN, AND M. DARUS

Assume that a3� �a22 is positive. Thus we now estimate Re. a3� �a22/, so from(2.8) and by using the Lemma 1 and letting p1 D 2rei� ; q1 D 2Rei� ;0 � r � 1;0�R � 1;0� � < 2� and 0� � < 2�; we obtain

3sC1Re.a3� �a22/

D�.�C1/

.�C2/.�C1/Re.q2�

1

2q21/C

Œ22sC1�.�C1/.�C1/�3sC1��2.�C2/�

22sC2.�C2/.�C1/2Req21

Cˇ�.�C1/

2.�C2/.�C1/Re.p2�

1

2p21/

CˇŒ22sC1�.�C1/.�C1/�3sC1��2.�C2/�

22sC1.�C2/.�C1/2Rep1q1

C3�22sˇ2�.�C1/.�C1/�3sC1�ˇ2�2.�C2/

22sC2.�C2/.�C1/2Rep21 (2.9)

�2�.�C1/

.�C2/.�C1/.1�R2/C

Œ22sC1�.�C1/.�C1/�3sC1��2.�C2/�

22s.�C2/.�C1/2R2 cos2�

Cˇ�.�C1/

.�C2/.�C1/.1� r2/

C2ˇ�22sC1�.�C1/.�C1/�3sC1��2.�C2/

�22s.�C2/.�C1/2

rRcos.�C�/

C3ˇ2Œ22s�.�C1/.�C1/�3s��2.�C2/�

22s.�C2/.�C1/2r2 cos2�

�3sC1��2.�C2/�22sC2�.�C1/.�C1/

22s.�C2/.�C1/2R2

C2ˇ�Œ3sC1��.�C2/�2sC1.�C1/.�C1/�

22s.�C2/.�C1/2rR

Cˇ

3ˇ�3s��2.�C2/�22s.�C1/.�C1/

�22s.�C2/.�C1/2

��.�C1/

.�C2/.�C1/

!r2

C�.�C1/

.�C2/.�C1/.ˇC2/

D .r;R/: (2.10)

Letting ˇ;�;� and � fixed and differentiating .r;R/ partially when �� 1;�� 0;ˇ � 1 and � � 1 we observe that

rrRR� .rR/2

D22sC4ˇ�2.�C1/2

.�C2/Œ2ˇC1��

4�3sC1�ˇ�3.�C1/

.�C1/Œ5ˇ�1� < 0:

COEFFICIENT INEQUALITY FOR CERTAIN CLASSES... 33

Therefore, the maximum of .r;R/ occurs on the boundaries. Thus the desiredinequality follows by observing that

.r;R/� .1;1/D3ˇ2

�3s��2.�C2/�22s�.�C1/.�C1/

�22s.�C2/.�C1/2

C

�.3sC1��2.�C2/�22sC1�.�C1/.�C1/

�.2ˇC1/

22s.�C2/.�C1/2:

The equality for (2.2) is attained when p1 D q1 D 2i and p2 D q2 D�2: �

Letting s D �D 0 and �D 1 in the above Theorem, we have the result given byJahangiri [9]:

Corollary 1. Let f .´/ 2K.ˇ/ and be given by (1.1). Then for ˇ � 1 and � � 1we have the sharp inequalityˇ

a3� �a22

ˇ� ˇ2.� �1/C

.2ˇC1/.3� �2/

3: (2.11)

Letting s D �D 1 and �D 0 in the above Theorem, we have the following result:

Corollary 2. Let f .´/ 2 T .1;0;1;ˇ/ and be given by (1.1). Then for ˇ � 1 and� � 1 we have the sharp inequalityˇ

a3� �a22

ˇ�1

36

�3ˇ2.3� �4/C .9� �8/.2ˇC1/

�:

ACKNOWLEDGEMENT

The authors would like to thank the referee for the informative comments andsuggestions in order to improve the manuscript.

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Authors’ addresses

T. Al-HawaryIrbid National University, Faculty of Science and Technology, Department of Mathematics, P.O.

Box: 2600, Irbid, JordanE-mail address: tariq amh@yahoo.com

B. A. FrasinFaculty of Science, Department of Mathematics, Al al-Bayt University, P.O. Box: 130095, Mafraq,

JordanE-mail address: bafrasin@yahoo.com

M. DarusSchool of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malay-

sia, Bangi 43600, Selangor D. Ehsan, MalaysiaE-mail address: maslina@ukm.edu.my