# Subclasses of Meromorphic Functions Associated with Convolution

- Maisarah Haji Mohd
^{1}, - Rosihan M. Ali
^{1}, - Lee See Keong
^{1}Email author and - V. Ravichandran
^{2}

**2009**:190291

**DOI: **10.1155/2009/190291

© Maisarah Haji Mohd et al. 2009

**Received: **12 December 2008

**Accepted: **11 March 2009

**Published: **16 March 2009

## Abstract

Several subclasses of meromorphic functions in the unit disk are introduced by means of convolution with a given fixed meromorphic function. Subjecting each convoluted-derived function in the class to be subordinated to a given normalized convex function with positive real part, these subclasses extend the classical subclasses of meromorphic starlikeness, convexity, close-to-convexity, and quasi-convexity. Class relations, as well as inclusion and convolution properties of these subclasses, are investigated.

## 1. Introduction

if there exists a Schwarz function , analytic in with and such that In particular, if the function is univalent in , then is equivalent to and

where is an analytic function with positive real part, , and maps the unit disk onto a region starlike with respect to 1.

In term of convolution, a function is starlike if is starlike, and convex if is starlike. These ideas led to the study of the class of all functions such that is starlike for some fixed function in In this direction, Shanmugam [2] introduced and investigated various subclasses of analytic functions by using the convex hull method [3–5] and the method of differential subordination. Ravichandran [6] introduced certain classes of analytic functions with respect to -ply symmetric points, conjugate points, and symmetric conjugate points, and also discussed their convolution properties. Some other related studies were also made in [7–9], and more recently by Shamani et al. [10].

Motivated by the investigation of Shanmugam [2], Ravichandran [6], and Ali et al. [7, 11], several subclasses of meromorphic functions defined by means of convolution with a given fixed meromorphic function are introduced in Section 2. These new subclasses extend the classical classes of meromorphic starlike, convex, close-to-convex, -convex, and quasi-convex functions given in (1.5). Section 3 is devoted to the investigation of the class relations as well as inclusion and convolution properties of these newly defined classes.

We will need the following definition and results to prove our main results.

Theorem 1.1 (see [12, Theorem 2.4]).

where denotes the closed convex hull of .

Theorem 1.2 (see [13]).

## 2. Definitions

In this section, various subclasses of are defined by means of convolution and subordination. Let be a fixed function in , and let be a convex univalent function with positive real part in and .

Definition 2.1.

Remark 2.2.

Definition 2.3.

The class consists of functions satisfying in and the subordination

Definition 2.4.

The class consists of functions such that in for some and satisfying the subordination

Definition 2.5.

For real, the class consists of functions satisfying , in and the subordination

Definition 2.6.

The class consists of functions such that in for some and satisfying the subordination

## 3. Main Results

This section is devoted to the investigation of class relations as well as inclusion and convolution properties of the new subclasses given in Section 2.

Theorem 3.1.

Let be a convex univalent function satisfying , and with . If , then . Equivalently, if , then .

Proof.

Theorem 3.2.

The function if and only if

Proof.

Theorem 3.3.

Let be a convex univalent function satisfying and with . If , then .

Proof.

hence .

Corollary 3.4.

*under the conditions of Theorem 3.3 .*

Proof.

The proof follows from (3.12).

In particular, when , the following corollary is obtained.

Corollary 3.5.

*Let*
*and*
*satisfy the conditions of Theorem 3.3 . If*
*, then*
.

Theorem 3.6.

Let and satisfy the conditions of Theorem 3.3. If , then . Equivalently .

Proof.

If , it follows from Theorem 3.2 that . Theorem 3.3 shows that Hence .

Theorem 3.7.

Under the conditions of Theorem 3.3, if with respect to , then with respect to .

Proof.

Theorem 3.3 shows that . Since , (3.9) yields .

Thus with respect to .

Corollary 3.8.

*under the assumptions of Theorem 3.3 .*

Proof.

The subordination (3.15) shows that .

Theorem 3.9.

Let . Then

(i) ,

(ii) for .

Proof.

- (i)Since and(3.19)

- (ii)Observe that(3.20)

Furthermore and from (i). Since and is convex, we deduce that . Therefore,

Corollary 3.10.

*The class*
*is a subset of the class*

Proof.

The proof follows from the definition of the classes by taking .

Theorem 3.11.

The function if and only if .

Proof.

Since , by Theorem 3.2 , . Hence .

Corollary 3.12.

*Let*
*and*
*satisfy the conditions of Theorem 3.3 . If*
, *then*

Proof.

If , Theorem 3.11 gives Theorem 3.7 next gives Thus, Theorem 3.11 yields

Corollary 3.13.

Proof.

If , it follows from Corollary 3.12 that The subordination

gives . Therefore

Open Problem

An analytic convex function in the unit disk is necessarily starlike. For the meromorphic case, is it true that ?

## Declarations

### Acknowledgment

The work presented here was supported in part by the USM's Reserach University grant and the FRGS grant

## Authors’ Affiliations

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