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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 192956, 14 pages

http://dx.doi.org/10.1155/2012/192956

## Fekete-Szegö Problems for Quasi-Subordination Classes

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Malaysia

Received 7 August 2012; Accepted 15 September 2012

Academic Editor: Juan J. Trujillo

Copyright © 2012 Maisarah Haji Mohd and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An analytic function is quasi-subordinate to an analytic function , in the open unit disk if there exist analytic functions and , with , and such that . Certain subclasses of analytic univalent functions associated with quasi-subordination are defined and the bounds for the Fekete-Szegö coefficient functional for functions belonging to these subclasses are derived.

#### 1. Introduction and Motivation

Let be the class of analytic function in the open unit disk normalized by and of the form . For two analytic functions and , the function is *subordinate* to , written as follows:
if there exists an analytic function , with and such that . In particular, if the function is univalent in , then is equivalent to and . For brief survey on the concept of subordination, see [1].

Ma and Minda [2] introduced the following class where is an analytic function with positive real part in , is symmetric with respect to the real axis and starlike with respect to and . A function is called Ma-Minda starlike (with respect to ). The class is the class of functions for which . The class and include several well-known subclasses of starlike and convex functions as special case.

In the year 1970, Robertson [3] introduced the concept of quasi-subordination. For two analytic functions and , the function is *quasi-subordinate* to , written as follows:
if there exist analytic functions and , with , and such that . Observe that when , then , so that in . Also notice that if , then and it is said that is *majorized* by and written in . Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization. See [4–6] for works related to quasi-subordination.

Throughout this paper it is assumed that is analytic in with . Motivated by [2, 3], we define the following classes.

*Definition 1.1. *Let the class consists of functions satisfying the quasi-subordination

*Example 1.2. *Since
the function defined by the following:
belongs to the class .

*Definition 1.3. *Let the class consists of functions satisfying the quasi-subordination

*Example 1.4. *The function defined by the following:
belongs to the class .

The classes and are analogous to the Ma-Minda starlike and convex classes defined in the form of quasi-subordination.

*Definition 1.5. *Let the class consist of functions satisfying the quasi-subordination

*Example 1.6. *The function defined by the following:
belongs to the class .

It is known that a function with in is univalent. The above class of functions defined in terms of the quasi-subordination is associated with the class of functions with positive real part.

Functions in the following classes, and are analogous to the -convex functions of Miller et al. [7] and -logarithmically convex functions introduced by Lewandowski et al. [8] (see also [9]), respectively.

*Definition 1.7. *Let the class , () consist of functions satisfying the quasi-subordination

*Example 1.8. *The function defined by the following:
belongs to the class .

*Definition 1.9. *Let the class , () consist of functions satisfying the quasi-subordination

*Example 1.10. *The function defined by the following:
belongs to the class .

It is well known (see [10]) that the -th coefficient of a univalent function is bounded by . The bounds for coefficient give information about various geometric properties of the function. Many authors have also investigated the bounds for the Fekete-Szegö coefficient for various classes [11–25]. In this paper, we obtain coefficient estimates for the functions in the above defined classes.

Let be the class of analytic functions , normalized by , and satisfying the condition . We need the following lemma to prove our results.

Lemma 1.11 (see [26]). *If , then for any complex number **
The result is sharp for the functions or . *

#### 2. Main Results

Although Theorems 2.1 and 2.4 are contained in the corresponding results for the classes and , they are stated and proved separately here because of the importance of the classes.

Throughout, let , , and .

Theorem 2.1. *If belongs to , then
**
and, for any complex number ,
*

*Proof. *If , then there exist analytic functions and , with , and such that

Since
it follows from (2.3) that
Since is analytic and bounded in , we have [27, page 172]
By using this fact and the well-known inequality, , we get
Further,
Then
Again applying and , we have
Applying Lemma 1.11 to
yields
Observe that
and hence we can conclude that
For , the above will reduce to the estimate of .

*Remark 2.2. *For , Theorem 2.1 gives a particular case of the estimates in [13, Theorem 1] for and [14, Theorem 2.1] for .

Theorem 2.3. *If satisfies
**
then the following inequalities hold:
**
and, for any complex number ,
*

*Proof. *The result follows by taking in the proof of Theorem 2.1.

Theorem 2.4. *If belongs to , then
**
and, for any complex number ,
*

*Proof. *Observe that when , equality (2.3) becomes
or equally
and the converse can be verified easily. By the Alexander relation, that is if and only if , we can obtain the required estimates.

Theorem 2.5. *If satisfies
**
then the following inequalities hold:
**
and, for any complex number ,
*

Theorem 2.6. *If belongs to , then
**
and, for any complex number ,
*

*Proof. *For , we know that by Definition 1.5 there exist analytic functions and , with , and such that
Since
it follows from (2.28) and (2.5) that
Following the same argument as in Theorem 2.1, where and , we can deduce that
Applying Lemma 1.11, we get
Since
and we can conclude the hypothesis.

Theorem 2.7. *If satisfies
**
then the following inequalities hold:
**
and, for any complex number ,
*

Let the class consist of functions satisfying the quasi-subordination where . The following corollary gives the results for .

Corollary 2.8. *Let . If belongs to , then
**
and, for any complex number ,
*

*Remark 2.9. *(1) For , Corollary 2.8 gives a particular case of the estimates in [13, Theorem 3] for and [14, Theorem 2.3] for .

(2) For and , , Corollary 2.8 reduces to the results in [19, Theorem 4].

Theorem 2.10. *Let . If belongs to , then
**
and, for any complex number ,
*

*Proof. *If , for then there are analytic functions and , with , and such that
A computation shows that
Hence from (2.43), we have
It then follows from relation (2.42) and (2.5) that
We can then conclude the proof by proceeding similarly as previous theorems.

*Remark 2.11. *(1) When , Theorem 2.10 reduces to Theorem 2.1.

(2) When , Theorem 2.10 reduces to Theorem 2.4.

(3) For , Theorem 2.10 gives a particular case of the estimates in [14, Theorem ] for .

Theorem 2.12. *Let . If satisfies
**
then the following inequalities hold:
**
and, for any complex number ,
*

Theorem 2.13. *Let and . If belongs to , then
**
and, for any complex number ,
*

*Proof. *If , for and then there are analytic functions and , with , and such that
A computation shows that
Thus (2.52) give
By using the above equation and (2.5) in (2.51) we have
We can proceed similarly as previous theorems and proof the hypothesis.

*Remark 2.14. *(1) When , Theorem 2.13 reduces to Theorem 2.4.

(2) When , Theorem 2.13 reduces to Theorem 2.1.

(3) For , Theorem 2.13 gives a particular case of the estimates in [14, Theorem ] for .

Theorem 2.15. *Let and . If satisfies
**
then the following inequalities hold:
**
and, for any complex number ,
*

#### Acknowledgment

The work presented here was supported in part by research Grant LRGS/TD/2011/UKM/ICT/03/02. The authors are thankful to the referees for their useful comments.

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