Journal of Applied Mathematics

Volume 2012 (2012), Article ID 161436, 9 pages

doi:10.1155/2012/161436

Research Article

Some Properties of Certain Integral Operators on New Subclasses of Analytic Functions with Complex Order

Aabed Mohammed and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 15 July 2012; Accepted 15 September 2012

Academic Editor: Roberto Natalini

Copyright © 2012 Aabed Mohammed and Maslina 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 define new subclasses of -valent meromorphic functions with complex order. We prove some properties for certain integral operators on these subclasses.

1. Introduction

Let be the open unit disc in the complex plane , , the punctured open unit disk. Let denote the class of meromorphic functions of the form

which are analytic and -valent in .

For , we obtain the class of meromorphic functions .

We say that a function is the meromorphic -valent starlike of order and belongs to the class , if it satisfies the inequality

A function is the meromorphic -valent convex function of order , if satisfies the following inequality:

and we denote this class by .

Many important properties and characteristics of various interesting subclasses of the class of meromorphically -valent functions were investigated extensively by (among others) Uralegaddi and Somanatha [1, 2], Liu and Srivastava [3, 4], Mogra [5, 6], Srivastava et a1. [7], Aouf et al. [8, 9], Joshi and Srivastava [10], Owa et a1. [11], and Kulkarni et al. [12].

Now, for , we define the following new subclasses.

Definition 1.1. Let a function be analytic in . Then is in the class if it satisfies the inequality

where

Definition 1.2. Let a function be analytic in . Then is in the class if it satisfies the inequality

where

Definition 1.3. Let a function be analytic in . Then is in the class if it satisfies the inequality

where

For in Definitions 1.1, 1.2, and 1.3, we obtain the following new subclasses of meromorphic functions .

Definition 1.4. Let a function be analytic in . Then is in the class if it satisfies the inequality

where

Definition 1.5. Let a function   be analytic in . Then is in the class if it satisfies the inequality

where

For in Definition 1.5, we obtain , the class of meromorphic function, introduced and studied by Wang et al. [13–15] (see [16–18]).

Definition 1.6. Let a function   be analytic in . Then is in the class if it satisfies the inequality

where

Most recently, Mohammed and Darus [19] introduced the following two general integral operators of -valent meromorphic functions :

where

For in (1.16) and (1.17), respectively, we obtain the general integral operators and , introduced by the authors in [16, 20].

2. Main Results

In this section, considering the above new subclasses we obtain for the integral operators and some sufficient conditions for a family of functions to be in the the above new subclasses.

Theorem 2.1. Let . If , then

where

Proof. A differentiation of which is defined in (1.16), we get

Then from , we obtain

By multiplying (2.4) with yield,

That is equivalent to

Equivalently, the above can be written as

Taking the real part of both terms of (2.7), we have

Sine , we get

That is,

Then

This completes the proof.

Theorem 2.2. Let . If , then

where

Proof. A differentiation of , which is defined in (1.17), we get

Then from , we obtain

That is equivalent to

Equivalently, the above can be written as

Taking the real part of both terms of (2.17), we have

Sine , we get

That is,

Then

This completes the proof.

Putting in Theorem 2.1, we have the following.

Theorem 2.3. Let . If , then

where

Putting in Theorem 2.2, we have the following.

Theorem 2.4. Let . If , then

where

for other work that we can look at regarding integral operators see [17, 21, 22].

Acknowledgment

This work was supported by UKM-ST-06-FRGS0244-2010 and LRGS/TD/2011/UKM/ICT/03/02.

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