International Journal of Mathematics and Mathematical SciencesVolume 2011 (2011), Article ID 459063, 11 pagesdoi:10.1155/2011/459063

Research Article

On Certain Class of Analytic Functions Related to Cho-Kwon-Srivastava Operator

F. Ghanim¹ and M. Darus²

¹Faculty of Management, Multimedia University, Selangor D. Ehsan, 63100 Cyberjaya, Malaysia²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor D. Ehsan, 43600 Bangi, Malaysia

Received 27 March 2011; Accepted 29 August 2011

Academic Editor: Stanisława R. Kanas

Copyright © 2011 F. Ghanim and M. 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.

Motivated by a multiplier transformation and some subclasses of meromorphic functions which were defined by means of the Hadamard product of the Cho-Kwon-Srivastava operator, we define here a similar transformation by means of the Ghanim and Darus operator. A class related to this transformation will be introduced and the properties will be discussed.

1. Introduction

Let denote the class of meromorphic functions normalized by which are analytic in the punctured unit disk . For , we denote by and the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order and convex of order in (cf. e.g., [1–4]).

For functions defined by we denote the Hadamard product (or convolution) of and by Let us define the function by for , and , where is the Pochhammer symbol. We note that where is the well-known Gaussian hypergeometric function.

Let us put Corresponding to the functions and and using the Hadamard product for , we define a new linear operator on by The meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava [5, 6], Liu [7], Liu and Srivastava [8–10], and Cho and Kim [11].

For a function , we define and, for , Note that if , , the operator reduced to the one introduced by Cho et al. [12] for . It was known that the definition of the operator was motivated essentially by the Choi-Saigo-Srivastava operator [13] for analytic functions, which includes a simpler integral operator studied earlier by Noor [14] and others (cf. [15–17]). Note also the operator has been recently introduced and studied by Ghanim and Darus [18] and Ghanim et al. [19], respectively. To our best knowledge, the recent work regarding operator was charmingly studied by Piejko and Sokól [20]. Moreover, the operator was then defined and studied by Ghanim and Darus [21]. In the same direction, we will study for the operator given in (1.10).

Now, it follows from (1.8) and (1.10) that Making use of the operator , we say that a function is in the class if it satisfies the following subordination condition: Furthermore, we say that a function is a subclass of the class of the form The main object of this paper is to present several inclusion relations and other properties of functions in the classes and which we have introduced here.

2. Main Results

We begin by recalling the following result (popularly known as Jack's Lemma), which we will apply in proving our first inclusion theorem.

Lemma 2.1 (see [Jack's Lemma] [22]). Let the (nonconstant) function be analytic in with . If attains its maximum value on the circle at a point , then where is a real number and .

Theorem 2.2. If then

Proof. Let , and suppose that where the function is either analytic or meromorphic in , with . By using (2.4) and (1.11), we have Upon differentiating both sides of (2.5) with respect to logarithmically and using the identity (1.11), we obtain We suppose now that and apply Jack's Lemma, we thus find that By writing and setting in (2.6), we find after some computations that Set Then, by hypothesis, we have which, together, imply that View of (2.13) and (2.10) would obviously contradict our hypothesis that Hence, we must have and we conclude from (2.4) that The proof of Theorem 2.2 is thus complete.

3. Properties of the Class

Throughout this section, we assume further that and We first determine a necessary and sufficient condition for a function of the form (1.13) to be in the class of meromorphically univalent functions with positive coefficients.

Theorem 3.1. Let be given by (1.13). Then if and only if where, for convenience, the result is sharp for the function given by for all .

Proof. Suppose that the function is given by (1.13) and is in the class . Then, from (1.13) and (1.12), we find that Since for any , therefore, we have Choosing to be real and letting through real values, (3.5) yields which leads us to the desired inequality (3.2).Conversely, by applying hypothesis (3.2), we get Hence, we have . By observing that the function , given by (3.3), is indeed an extremal function for the assertion (3.2), we complete the proof of Theorem 3.1.

By applying Theorem 3.1, we obtain the following sharp coefficient estimates.

Corollary 3.2. Let be given by (1.13). If , then where the equality holds true for the function given by (3.3).

Next, we prove the following growth and distortion properties for the class .

Theorem 3.3. If the function defined by (1.13) is in the class , then, for , we have Each of these results is sharp with the extremal function given by (3.3).

Proof. Since . Theorem 3.1 readily yields the inequality Thus, for and utilizing (3.10), we have Also from Theorem 3.1, we get Hence This completes the proof of Theorem 3.3.

We conclude this section by determining the radii of meromorphically univalent starlikeness and meromorphically univalent convexity of the class . We state our results as in the following theorems.

Theorem 3.4. Let . Then, is meromorphically univalent starlike of order in , where The equality is attained for the function given by (3.3).

Proof. It suffices to prove that for , we have Hence, (3.16) holds true if or with the aid of (3.18) and (3.2), it is true to have Solving (3.19) for , we obtain This completes the proof of Theorem 3.4.

Theorem 3.5. Let . Then, is meromorphically univalent convex of order in , where The equality is attained for the function given by (3.3).

Proof. By using the technique employed in the proof of Theorem 3.4, we can show that for , with the aid of Theorem 3.1. Thus, we have the assertion of Theorem 3.5.

Acknowledgment

The work presented here was fully supported by UKM-ST-06-FRGS0244-2010.

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