- Research Article
- Open Access

# 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

https://doi.org/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.

## Keywords

- Analytic Function
- Unit Disk
- Meromorphic Function
- Conjugate Point
- Class Relation

## 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.

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

## References

- Ma WC, Minda D:
**A unified treatment of some special classes of univalent functions.**In*Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 1994, Cambridge, Mass, USA, Conference Proceedings and Lecture Notes in Analysis, I*. International Press; 157–169.MATHGoogle Scholar - Shanmugam TN:
**Convolution and differential subordination.***International Journal of Mathematics and Mathematical Sciences*1989,**12**(2):333–340. 10.1155/S0161171289000384MathSciNetView ArticleMATHGoogle Scholar - Barnard RW, Kellogg C:
**Applications of convolution operators to problems in univalent function theory.***The Michigan Mathematical Journal*1980,**27**(1):81–94.MathSciNetView ArticleMATHGoogle Scholar - Parvatham R, Radha S:
**On -starlike and -close-to-convex functions with respect to -symmetric points.***Indian Journal of Pure and Applied Mathematics*1986,**17**(9):1114–1122.MathSciNetMATHGoogle Scholar - Ruscheweyh S, Sheil-Small T:
**Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture.***Commentarii Mathematici Helvetici*1973,**48:**119–135. 10.1007/BF02566116MathSciNetView ArticleMATHGoogle Scholar - Ravichandran V:
**Functions starlike with respect to -ply symmetric, conjugate and symmetric conjugate points.***The Journal of the Indian Academy of Mathematics*2004,**26**(1):35–45.MathSciNetMATHGoogle Scholar - Ali RM, Ravichandran V, Lee SK: Subclasses of multivalent starlike and convex functions. to appear in Bulletin of the Belgian Mathematical Society. Simon Stevin to appear in Bulletin of the Belgian Mathematical Society. Simon StevinGoogle Scholar
- Padmanabhan KS, Parvatham R:
**Some applications of differential subordination.***Bulletin of the Australian Mathematical Society*1985,**32**(3):321–330. 10.1017/S0004972700002410MathSciNetView ArticleMATHGoogle Scholar - Padmanabhan KS, Parvatham R:
**Some applications of differential subordination.***Bulletin of the Australian Mathematical Society*1985,**32**(3):321–330. 10.1017/S0004972700002410MathSciNetView ArticleMATHGoogle Scholar - Shamani S, Ali RM, Ravichandran V, Lee SK: Convolution and differential subordination for multivalent functions. to appear in Bulletin of the Malaysian Mathematical Sciences Society. Second Series to appear in Bulletin of the Malaysian Mathematical Sciences Society. Second SeriesGoogle Scholar
- Ali RM, Ravichandran V, Seenivasagan N:
**Subordination and superordination of the Liu-Srivastava linear operator on meromorphic functions.***Bulletin of the Malaysian Mathematical Sciences Society. Second Series*2008,**31**(2):193–207.MathSciNetMATHGoogle Scholar - Ruscheweyh S:
*Convolutions in Geometric Function Theory: Fundamental Theories of Physics, Séminaire de Mathématiques Supérieures*.*Volume 83*. Presses de l'Université de Montréal, Montreal, Canada; 1982:168.Google Scholar - Eenigenburg P, Miller SS, Mocanu PT, Reade MO:
**On a Briot-Bouquet differential subordination.***Revue Roumaine de Mathématiques Pures et Appliquées*1984,**29**(7):567–573.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.