ISRN Applied MathematicsVolume 2012 (2012), Article ID 272678, 11 pagesdoi:10.5402/2012/272678

Research Article

On a Class of Meromorphic Functions Related to Cho-Kwon-Srivastava Operator

F. Ghanim¹ and M. Darus²

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

Received 2 January 2012; Accepted 13 February 2012

Academic Editors: X.-S. Yang and W. Yeih

Copyright © 2012 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.

We introduce new classes, and of meromorphic functions defined by means of the Hadamard product of Cho-Kwon-Srivastava operator, and we define here a similar transformation by means of an operator given by Ghanim and Darus, in and investigate a number of inclusion relationships of these classes. We also derive some interesting properties of these classes.

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 .

For functions defined by we denote the Hadamard product (or convolution) of and by

Let be the class of functions analytic in satisfying the properties and where , and . This class has been introduced in [1]. We note that and , see ([2, 3]), the class of analytic functions with positive real part greater than and , the class of functions with positive real part. From (1.6), we can write as where and .

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 [4, 5], Liu [6], Liu and Srivastava [7–9], and Cho and Kim [10].

For a function , we define and for , Note that if , the operator has been introduced by Cho et al. [11] for . It was known that the definition of the operator was motivated essentially by the Choi-Saigo-Srivastava operator [12] for analytic functions, which includes a simpler integral operator studied earlier by Noor [13] and others (cf. [14–16]). Note also the operator has been recently introduced and studied by Ghanim and Darus, [17], [18] and [19], respectively. To our best knowledge, the recent work regarding operator was charmingly studied by Piejko and Sokół [20]. Moreover, the operator was defined and studied by Ghanim and Darus [21]. In the same direction, we will study the operator given in (1.12).

Now, it follows from (1.10) and (1.12) that

In the present paper, we will use of the operator and introduce some new classes of meromorphic functions.

Definition 1.1. Let . Then if and only if where , and .

Definition 1.2. Let . Then if and only if where , and .

2. Preliminary Results

Lemma 2.1 (see [22]). If is analytic in with , and if is a complex number satisfying , then implies where is given by which is an increasing function of and . The estimate is sharp in the sense that the bound cannot be improved.

Lemma 2.2 (see [23]). If is analytic in , , and , then for any function analytic in , the function takes values in the convex hull of the image of under .

Lemma 2.3 (see [24]). Let . Then

3. Main Results

Theorem 3.1. Let . Then where is given by

Proof. Let Then is analytic in with . Applying the identity (1.13) in (3.4) and differentiating the resulting equation with respect to , we have Since , so for . This implies that Using Lemma 2.1, we see that , where is given by (3.2). Consequently, for , and the proof is complete.

Theorem 3.2. Let for . Then for , where

Proof. Set Now proceeding as in Theorem 3.1, we have where we have used (1.6) and . Using the following well-known estimate [25] we have The right-hand side of this inequality is positive if , where is given by (3.7). Consequently, it follows from (3.9) that for . Sharpness of this result follows by taking in (3.9), .

Theorem 3.3. Let , and let Then where is given by

Proof. Setting then is analytic in with . Using the following operator identity: in (3.16), and differentiating the resulting equation with respect to , we find that Using Lemma 2.1, we see that for , where is given by (3.14).Hence, the proof is complete.

Theorem 3.4. Let satisfy the following inequality: If , then . Proof. Let . Then Since and by using Lemma 2.2, we can conclude that .

Theorem 3.5. Let satisfy the inequality (3.19) and . Then .

Proof. We have Now the remaining part of Theorem 3.5 follows by employing the techniques that we used in proving Theorem 3.4 above.

Theorem 3.6. Let and , and let . Then , where This result is sharp.

Proof. Since and , it follows that and so using identity (1.13) in the above equation, we have Using (3.24), we have where Now where and . Since we obtain that , by using the Herglots formula. Thus with Using (3.25), (3.26), (3.28), (3.31), and Lemma 2.3, we have From this, we conclude that , where is given by (3.22). We discuss the sharpness as follows: we take Since it follows from (3.26) that as . This completes the proof.

Note 1. Some other works related to Cho-Kwon-Srivastava operator can also be referred to [26–29].

Acknowledgments

The work presented here was fully supported by UKM-ST-06-FRGS0244-2010. The authors also would like to thank the referees for some suggestions to improve the content of the paper.

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