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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 638251, 8 pages
http://dx.doi.org/10.1155/2008/638251
Research Article

Subordination Properties for Certain Analytic Functions

1Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura 35516, Egypt
2Department of Mathematics, Teachers' College in Abha, King Khalid University, Abha, P.O. Box 249, , Saudi Arabia
3Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea
4Department of Mathematics, Faculty of Science, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 13 September 2007; Accepted 7 November 2007

Academic Editor: Brigitte Forster-Heinlein

Copyright © 2008 A. A. Attiya 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 purpose of the present paper is to derive a subordination result for functions in the class of normalized analytic functions in the open unit disk . A number of interesting applications of the subordination result are also considered.

1. Introduction

Let denote the class of functions of the form which are analytic in the unit disc . We also denote by the class of functions that are convex in .

Given two functions where is given by (1.1) and is defined by the Hadamard product (or convolution) is defined by

By using the Hadamard product, Ruscheweyh [1] defined From the definition of (1.4), we observe that when . The symbol () was called the -th order Ruscheweyh derivative of by Al-Amiri [2]. We also note that and .

Definition 1.1. Suppose that . Then the function is said to be a member of the class if it satisfies By specializing , , , and , one can obtain various subclasses studied by many authors (see, e.g., [311]).

Definition 1.2. Let be analytic and univalent in . If is analytic in , , and then one says that is subordinate to in , and one writes or . One also says that is superordinate to in .

Definition 1.3. An infinite sequence of complex numbers will be called a subordinating factor sequence if for every univalent function in , one has

Lemma 1.4. (see [12]). The sequence is a subordinating factor sequence if and only if

Now, we prove the following lemma which gives a sufficient condition for functions belonging to the class

Lemma 1.5. If the function which is defined by (1.1) satisfies the following condition: where then .

Proof. Suppose that the inequality (1.9) holds. Using the identity we have for , which shows that belongs to .

Let denote the class of functions in whose Taylor-Maclaurin coefficients satisfy the condition (1.9).

We note that

Example 1.6. (i) For , , , and , the following function defined by: is in the class
(ii) For , , , and , the following functions defined by: are in the class

In this paper, we obtain a sharp subordination result associated with the class by using the same techniques as in [13] (see also [1416]). Some applications of the main result which give important results of analytic functions are also investigated.

2. Main Theorem

Theorem 2.1. Let . Then for every function in , and The constant cannot be replaced by a larger one.

Proof. Let and let be any function in the class . Then we readily have Thus, by Definition 1.2, the subordination result (2.1) will hold true if the sequence is a subordinating factor sequence, with In view of Lemma 1.4, this is equivalent to the following inequality: Now, since is an increasing function of , we have This proves the inequality (2.6), and hence also the subordination result (2.1) asserted by Theorem 2.1. The inequality (2.2) follows from (2.1) by taking
Next, we consider the function which is a member of the class Then by using (2.1), we have It can be easily verified for the function defined by (2.10) that which completes the proof of Theorem 2.1.

3. Some Applications

Taking in Theorem 2.1, we obtain the following.

Corollary 3.1. If the function defined by (1.1) satisfies then for every function in , one has The constant cannot be replaced by larger one.

Putting in Theorem 2.1, we have the following corollary.

Corollary 3.2. If the function defined by (1.1) satisfies where is defined by (1.10), then for every function in , one has The constant cannot be replaced by larger one.

Next letting and in Theorem 2.1, we obtain the following corollary.

Corollary 3.3. If the function satisfies then for every function in , one has The constant cannot be replaced by a larger one.

Remark 3.4. Putting , , and , in Theorem 2.1, we obtain the result due to Singh [17].

Also, by taking and in Theorem 2.1, we have the following.

Corollary 3.5. If the function satisfies then for every function in , one has The constant cannot be replaced by a larger one.

It is clearly from the proof of Theorem 2.1 that the function () is the extremal function of Corollary 3.5. Also, the following example gives a nonpolynomial extremal function for the same corollary.

Example 3.6. Let the function be defined by the above function is analytic in and it is equivalent to Then we have Therefore, the Taylor-Maclaurin coefficients of the function satisfy the condition in Corollary 3.5. Moreover, it can be easily verified that Then, the constant cannot be replaced by a larger one. Therefore, the function is the extremal function of Corollary 3.5.

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