Journal of Applied Mathematics

Volume 2013 (2013), Article ID 127615, 7 pages

http://dx.doi.org/10.1155/2013/127615

Research Article

Fekete-Szegő Inequality for a Subclass of -Valent Analytic Functions

Mohsan Raza,¹ Muhammad Arif,² and Maslina Darus³

¹Department of Mathematics, Government College University Faisalabad, Faisalabad, Punjab 38000, Pakistan

²Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa 23200, Pakistan

³School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor D. Ehsan, Malaysia

Received 18 January 2013; Accepted 11 April 2013

Academic Editor: Mina Abd-El-Malek

Copyright © 2013 Mohsan Raza et al. 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.

The main object of this paper is to study Fekete-Szegő problem for the class of -valent functions. Fekete-Szegő inequality of several classes is obtained as special cases from our results. Applications of the results are also obtained on the class defined by convolution.

1. Introduction and Preliminaries

Let denote the class of functions of the form

which are analytic in the open unit disk . Also, , the usual class of analytic functions defined in the open unit disk . Let and be analytic in . We say that the function is subordinate to the function and write , if and only if there exists Schwarz function , analytic in such that , for , and . In particular, if is univalent in , then we have the following equivalence:

For any two analytic functions of the form (1) and with

the convolution (Hadamard product) is given by

Let be an analytic function with positive real part on with , , which maps the unit disk onto a region starlike with respect to which is symmetric with respect to the real axis. Denote by the class of functions analytic in for which

The class was defined and studied by Ali et al. [1]. They obtained the Fekete-Szegő inequality for functions in the class . The class coincides with the class discussed by Ma and Minda [2]. Owa [3] introduced a subclass of -valently Bazilevic functions . A function is said to be in the class if and only if

where , , and . We now define the following subclass of analytic functions.

Definition 1. Let be a univalent starlike function with respect to 1 which maps the unit disk onto a region in the right half-plane which is symmetric with respect to the real axis with and . A function is in the class if

where , , and .

Definition 2. A function is in the class if

where , , and In other words, a function is in the class if .

We have the following special cases.

(i) coincides with the class  introduced and studied by Ali et al. [1].(ii)For , , and , we have the class introduced and studied by Ravichandran et al. [4].(iii)For and , the class reduces to introduced and studied by Owa [3].(iv)For , the class reduces to the class defined as

(v)For , the class is defined as

(vi) is investigated by Ma and Minda [2].(vii)For , the class reduces to the class

studied by Chen [5].

We need the following results to obtain our main results.

Lemma 3 (see [1]). Let be the class of analytic functions normalized by satisfying condition If and , then

For or the equality holds, if and only if or one of its rotation. For the equality holds, if or one of its rotation. the equality holds for , if and only if or one of its rotation, while for , the equality holds, if and only if or one of its rotation. The above upper bound for is sharp, and it can be improved as follows:

Lemma 4 (see [6, (7), page 10]). If and , then

for any complex number . The result is sharp for the functions  or .

Lemma 5 (see [7]). If , then for any real number and , the following sharp estimate holds:

where

The extremal function up to the rotations is of the form

The sets , are defined as follows:

2. Main Results

Theorem 6. Let , where , are real with and . Let

If is of the form (1) and belongs to the class , then

Furthermore, for ,

and for ,

For any complex number ,

Also,

where is defined in Lemma 3 and

These results are sharp.

Proof. Since , therefore we have for a Schwarz function

such that

Now,

Also, we have

Comparing the coefficients of , , and after simple calculations, we obtain

where and are defined in (25). It can be easily followed from (30) that

where

The results from (20) to (22)are obtained by using Lemma 3, (23)by using Lemma 4, and (24) by using Lemma 5. To show that these results are sharp, we define the functions , , and such that

with

It is clear that the functions . Let . If or , then the equality occurs for the function or one of its rotations. For , the equality is attained, if and only if is or one of its rotations. When , then the equality holds for the function or one of its rotations. If , then the equality is obtained for the function or one of its rotations.

Corollary 7. For , the results from (20) to (24) coincide with the results proved by Ramachandran et al. [8].

Corollary 8. For ,  , and the results from (20) to (22) coincide with the results obtained by Ravichandran et al. [4] for the class .

Corollary 9. For ,  , and the results from (20) to (24) coincide with the results obtained by Ali et al. [1] for the class .

Corollary 10. For , , , and the results from (20) to (22) coincide with the results obtained by Ma and Minda [2] for the class .

2.1. Application of Theorem 6 to the Function Defined by Convolutions

Theorem 11. Let , where , are real with and . Let

If is of the form (1) and belongs to the class , then

Furthermore, for ,

and for ,

For any complex number ,

Also,

where is defined in Lemma 5 and

These results are sharp.

Proof. Since , therefore we have for a Schwarz function such that

Now,

Also, we obtain

Comparing the coefficients of , , and after simple calculations, we obtain

The remaining proof of the theorem is similar to the proof of Theorem 6.

Corollary 12. For , the results from (36) to (40) coincide with the results proved by Ramachandran et al. [8] for the class .

Corollary 13. For , , and the results from (36) to (40) coincide with the results obtained by Ali et al. [1] for the class .

Corollary 14. For , , , and ,

the results from (36) to (38) coincide with the results obtained by Srivastava and Mishra [9].

Acknowledgment

The work here is fully supported by LRGS/TD/2011/UKM/ICT/03/02.

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