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Journal of Complex Analysis
Volume 2013 (2013), Article ID 851318, 4 pages
http://dx.doi.org/10.1155/2013/851318
Research Article

Certain Subclasses of -Valent Meromorphic Functions Associated with a New Operator

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 21 December 2012; Accepted 10 March 2013

Academic Editor: J. Dziok

Copyright © 2013 A. O. Mostafa 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 introduce two classes of -valent meromorphic functions associated with a new operator and derive several interesting results for these classes.

1. Introduction

Let denote the class of functions of the form which are analytic and -valent in the punctured unit disc . Let be the class of functions analytic in satisfying the properties and where , , and . The class was introduced by Padmanabhan and Parvatham [1]. For , the class was introduced by Pinchuk [2]. Also we note that , where is the class of functions with positive real part greater than and , where is the class of functions with positive real part. From (2), we have if and only if there exists such that It is known that the class is a convex set (see [3]).

For functions given by (1) and given by the Hadamard product (or convolution) of and is defined by

Aqlan et al. [4] defined the operator by

Mostafa [5] used Aqlan et al. operator and defined the following linear operator as follows.

First put and let be defined by Then Using (7) and (9), we have where denotes the Pochhammer symbol given by It is readily verified from (10) that (see [5])

It is noticed that by putting in (10), we obtain the operator

Now, by using the linear operator , we introduce classes of -valent Bazilevic functions of as follows.

Definition 1. A function is said to be in the class if it satisfies the following condition:

Definition 2. A function is said to be in the class if it satisfies the following condition:

In this paper, we investigate several properties of the classes and .

2. Main Results

Unless otherwise mentioned, we assume throughout this paper that , , , , , and all powers are understood as principle values.

To prove our results we need the following lemma.

Lemma 3 (see [6]). Let , and be a complex-valued function satisfying the conditions:(i) is continuous in a domain .(ii) and .(iii) whenever and .
If is analytic in such that and for , then in .

Employing the technique used by Noor and Muhammad [7] and Aouf and Seoudy [8] for multivalent functions, we prove the following theorems.

Theorem 4. If , then where is given by

Proof. Setting where are analytic in with , and is given by (3). Differentiating both sides of (19) with respect to and using (12) in the resulting equation, we obtain which implies that We form the functional by choosing , , that is, Clearly, the first two conditions of Lemma 3 are satisfied. Now, we verify the condition (iii) as follows: where We note that if and only if , . From given by (18), we have , , and . Therefore, applying Lemma 3, we have and consequently for . This completes the proof of Theorem 4.

Similarly, we can prove the following theorem for the class .

Theorem 5. If , then where is given by

Theorem 6. If , then where is given by

Proof. Let and where are analytic in with and is given by (3). Differentiating both sides of (29) with respect to and using (12) in the resulting equation, we obtain which implies that We form the functional by choosing , , that is, Clearly, the conditions (i) and (ii) of Lemma 3 are satisfied. Now, we verify the condition (iii) as follows: where We note that if and only if , . From as given by (28), we have , , and . Therefore, applying Lemma 3, we have and consequently for . This completes the proof of Theorem 6.

Similarly, we can prove the following theorem for the class .

Theorem 7. If , then where is given by

Remark 8. Putting , in Theorems 4 and 6, we obtain the corresponding results for the operator defined in (14).

Acknowledgment

The authors would like to thank the referees of the paper for their helpful suggestions.

References

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