ISRN GeometryVolume 2011 (2011), Article ID 812541, 13 pagesdoi:10.5402/2011/812541

Research Article

Inclusion Relationships for Certain Subclasses of Meromorphic Functions Defined by Using the Extended Multiplier Transformations

R. M. El-Ashwah

Department of Mathematics, Faculty of Science (Damietta Branch), Mansoura University, New Damietta 34517, Egypt

Received 22 May 2011; Accepted 4 July 2011

Academic Editors: A. Belhaj and A. Masiello

Copyright © 2011 R. M. El-Ashwah. 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.

Let denote the class of analytic functions in the punctured unit disc . Set , and define in terms of the Hadamard product by . In this paper, we introduce several new subclasses of analytic functions defined by means of the operator Inclusion properties of these classes and some applications involving integral operator are also considered.

1. Introduction

Let denote the class of functions of the form: which are analytic in the punctured open unit disk and . We denote by , and the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order in , convex of order in , and close-to-convex of order and type in (see [1–3]).

Let be the class of all function which are analytic and univalent in and for which is convex with

For two functions and analytic in , we say that is subordinate to and write in or , if there exists a Schwarz function , which is analytic in with and , such that . It is known that Furthermore, if the function is univalent in (see, [4, page 4]),

Making use of the principle of subordination between analytic functions, we define the subclasses , and of the class for and , which are defined by respectively. For special choices for the parameters and as well as for special choices for the function and , we will obtain various subclasses of meromorphic function of the above classes (see [5–7]).

For , we define the multiplier transformation for functions (see [8, 9] with ) by Obviously, we have for all integers and .

We note that(i)(see [10, 11]);(ii) (see [12]);(iii)(see [13]).

Setting we define a new function in terms of the Hadamard product (or convolution) by Essentially Choi et al. [14] motivated the Choi-Saigo-Srivastava operator for analytic functions, which includes an integral operator considered earlier by Noor [15] and others [16–18]; we now introduce the operator which is defined here by; We note that(i)and ;(ii) (see [11]).

It is easily verified from the definition of the operator that

Next, by using the operator defined by (1.10), we introduce the following subclasses of meromorphic functions: We also note that In particular, we set

The main object of this paper is to investigate several inclusion properties of the classes mentioned above. Some applications involving integral operator are also considered.

2. Inclusion Properties Involving the Operator

The following lemmas will be required in our investigation.

Lemma 2.1 (see [19]). Let be convex univalent in with and . If is analytic in with , then implies that

Lemma 2.2 (see [20]). Let be convex univalent in and let be analytic in with . If is analytic in and , then implies that

At first, with the help of Lemma 2.1, we prove the following theorem.

Theorem 2.3. Let with then

Proof. We begin by showing the first inclusion relationship: which is asserted by Theorem 2.3. Let and set where the function is analytic in with . Then, by applying (1.11) is (2.8), we obtain Differentiating (2.9) logarithmically with respect to and multiplying the resulting equation by , we have Since we see that Applying Lemma 2.1 to (2.10), it follows that in , that is, For the second inclusion relationship asserted by Theorem 2.3, using arguments similar to those detailed above with (1.11), we obtain We thus complete the proof of Theorem 2.3.

Theorem 2.4. Let with then

Proof. Applying (1.14) and Theorem 2.3, we observe that which evidently prove Theorem 2.4.

By setting in Theorems 2.3 and 2.4, we deduce the following corollary.

Corollary 2.5. Suppose that Then, for the function classes defined by (1.15),

Next by using Lemma 2.2, one obtains the following inclusion relationships for the class .

Theorem 2.6. Let with Then

Proof. We begin by proving that which is the first inclusion relationship asserted by Theorem 2.6. Let Then, in view of the definition of the function class , there exists a function such that Choose the function such that . Then and Now let where the function is analytic in with . Using (1.12), we find that Since by Theorem 2.3, then we set where in with the assumption that . Then by (2.27) and (2.28), we observe that Differentiating both sides of (2.31) with respect to , multiplying by and dividing by , we obtain Now making use of (2.26), (2.32), and (2.33), we get Since and in with we have Hence, by taking in (2.34), and then applying Lemma 2.2, we can show that in , so that For the second inclusion relationship asserted by Theorem 2.6, using arguments similar to those detailed above with (1.11), we obtain We thus complete the proof of Theorem 2.6.

3. Inclusion Properties Involving the Integral Operator

In this section, we consider the integral operator (see, [4, page 11]) defined by From the definition (3.1), it is easily verified that By using (3.2) we can prove the following theorems (see Cho et al. [11]).

Theorem 3.1. Let with If then

Theorem 3.2. Let with If , then

From Theorems 3.1 and 3.2, we can easily deduce the following.

Corollary 3.3. Suppose that Then for the function classes defined by (1.15), the following inclusion relationships hold true:

Theorem 3.4. Let with If , then

Remark 3.5. Putting in all the above results, we will obtain the results obtained by Cho et al. [11].

Acknowledgment

The author thanks the referees for their valuable suggestions which led to improvement of this paper.

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