Abstract and Applied AnalysisVolume 2011 (2011), Article ID 375897, 8 pagesdoi:10.1155/2011/375897

Research Article

Certain Subordination Properties for Subclasses of Analytic Functions Involving Complex Order

S. Sivasubramanian,¹ Aabed Mohammed,² and Maslina Darus²

¹Department of Mathematics, University College of Engineering, Anna University of Technology Chennai, Melpakkam 604 001, India²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia

Received 25 June 2011; Accepted 14 October 2011

Academic Editor: Jing Ping Wang

Copyright © 2011 S. Sivasubramanian 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.

We derive several subordination results for a certain class of analytic functions defined by the Sălăgean operator In the present investigation.

1. Introduction and Preliminaries

Let denote the class of functions of the form which are analytic in the open unit disk Further, by we will denote the class of all functions in which are univalent in .

Also let denote, respectively, the subclasses of consisting of functions that are starlike of complex order , convex of complex order in . In particular, the classes and are the familiar classes of starlike and convex functions in .

Slgean [1] introduced the following operator which is popularly known as the Slgean derivative operator: and, in general, It is easy to see that from (1.1), Let denote the subclass of consisting of functions which satisfy Equivalently, We note that, for ,(1), (2), (3), (4), (5), (6), (7).

The class was studied by Abdul Halim [2], while the class was studied by Chen [3, 4] and the class was studied by Èzrohi [5] (see also the works of Altintas and Özkan [6], Aouf et al. [7], Attiya [8], Kamali and Akbulut [9], Özkan [10], and Shanmugam et al. [11]). A systematic investigation of the starlike and convex functions involving Slgean derivative was done by Aouf et al. very recently [7].

In our proposed investigation of functions in these subclasses of the normalized analytic function class , we need the following definitions and results.

Definition 1.1 (Hadamard product or convolution). For functions and in the class , where of the form (1.1) and is given by the Hadamard product (or convolution) is defined by

Definition 1.2 (subordination principle). For analytic functions and with is said to be subordinate to , denoted by , if there exists an analytic function such that for all .

Definition 1.3 (see [12], subordinating factor sequence). A sequence of complex numbers is said to be a subordinating sequence if, whenever is of the form (1.1) is analytic, univalent, and convex in , one has the subordination given by

Lemma 1.4 (see [12]). The sequence is a subordinating factor sequence for the class of convex univalent functions if and only if

2. Main Results

Theorem 2.1. Let the function of the form (1.1) satisfy the following condition: Then .

Proof. Suppose the inequality (2.1) holds. Then we have for which shows that belongs to the class .

In view of Theorem 2.1, we now introduce the subclass which consists of functions whose Taylor-Maclaurin coefficients satisfy the inequality (2.1). We note that .

In this work, we prove several subordination relationships involving the function class employing the technique used earlier by Attiya [13] and Srivastava and Attiya [14].

Theorem 2.2. Let , and let be any function in the usual class of convex functions , then for every function . Further, The constant factor in (2.3) cannot be replaced by a larger number.

Proof. Let , and suppose that . Then Thus, by Definition 1.3, the subordination result holds true if is a subordinating factor sequence, with . In view of Lemma 1.4, this is equivalent to the following inequality: Since is an increasing function of , we have, for , where we have also made use of the assertion (2.1) of Theorem 2.1. This evidently proves the inequality (2.3) and hence also the subordination result (2.3) asserted by Theorem 2.2. The inequality (2.4) follows from (2.3) by taking To prove the sharpness of the constant , we consider the function defined by Thus, from (2.3), we have It is easily verified that This shows that the constant cannot be replaced by any larger one.

For the choices of and , we get the following corollary.

Corollary 2.3. Let let and be any function in the usual class of convex functions , then where and , The constant factor in (2.13) cannot be replaced by a larger number.

For the choices of and , one gets the following.

Corollary 2.4. Let , and let be any function in the usual class of convex functions , then where and , The constant factor in (2.15) cannot be replaced by a larger number.

For the choices of , and , one gets the following.

Corollary 2.5. Let , and let be any function in the usual class of convex functions , then where and , The constant factor in (2.17) cannot be replaced by a larger number.

For the choices of , and , one gets the following.

Corollary 2.6. Let , and let be any function in the usual class of convex functions , then where , The constant factor in (2.19) cannot be replaced by a larger number.

For the choices of , and , one gets the following.

Corollary 2.7. Let , and let be any function in the usual class of convex functions , then where , The constant factor in (2.21) cannot be replaced by a larger number.

Acknowledgment

The authors thank the referees for their insightful suggestions in the paper. The second and third authors were partially supported by MOHE: UKM-ST-06-FRGS0244-2010.

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