# Convolution Properties of Classes of Analytic and Meromorphic Functions

- RosihanM Ali
^{1}Email author, - MoradiNargesi Mahnaz
^{1}, - V Ravichandran
^{2}and - KG Subramanian
^{1}

**2010**:385728

https://doi.org/10.1155/2010/385728

© Rosihan M. Ali et al. 2010

**Received: **30 October 2009

**Accepted: **13 May 2010

**Published: **13 June 2010

## Abstract

General classes of analytic functions defined by convolution with a fixed analytic function are introduced. Convolution properties of these classes which include the classical classes of starlike, convex, close-to-convex, and quasiconvex analytic functions are investigated. These classes are shown to be closed under convolution with prestarlike functions and the Bernardi-Libera integral operator. Similar results are also obtained for the classes consisting of meromorphic functions in the punctured unit disk.

## 1. Motivation and Definitions

Let be the set of all analytic functions defined in the unit disk . Denote by the class of normalized analytic functions defined in . For two functions and in , the convolution or Hadamard product of and is the function defined by . A function is subordinate to an analytic function , written , if there exists a Schwarz function , analytic in with and satisfying If the function is univalent in , then is equivalent to and .

By adding the two inequalities, it is evident that the function is starlike and hence both and are close-to-convex and univalent. This motivates us to consider the following classes of functions.

It is assumed in the sequel that is a fixed integer, is a fixed function in , and is a convex univalent function with positive real part in satisfying .

Definition 1.1.

The class consists of for which .

which shows that the function is starlike in . Thus, it follows from (1.4) that the component function of is close-to-convex in , and hence univalent. Similarly, the component function of is univalent.

then the class coincides with the class studied in [2], which there was denoted by , and reduces to a class introduced in [3] which there was denoted by . It is evident that the classes and extend the classical classes of starlike and convex functions, respectively.

Definition 1.2.

for some with , . In this case, we say that with respect to . The class consists of for which .

When , the classes and reduce, respectively, to and introduced and investigated in [1]. If , where is defined by (1.8), then the class coincides with studied in [2]. Clearly the classes and extend the classical classes of close-to-convex and quasiconvex functions, respectively.

while consists of satisfying .

The well-known result that the classes of starlike functions of order and convex functions of order are closed under convolution with prestarlike functions of order follows from the following.

Theorem 1.3 (see [4, Theorem ]).

for any analytic function , where denotes the closed convex hull of .

In the following section, by using the methods of convex hull and differential subordination, convolution properties of functions belonging to the four classes , , and , are investigated. It would be evident that various earlier works, see, for example, [5–10], are special instances of our work.

In Section 3, new subclasses of meromorphic functions are introduced. These subclasses extend the classical subclasses of meromorphic starlike, convex, close-to-convex, and quasiconvex functions. Convolution properties of these newly defined subclasses will be investigated. Simple consequences of the results obtained will include the work of Bharati and Rajagopal [6] involving the function , , as well as the work of Al-Oboudi and Al-Zkeri [5] on the modified Salagean operator.

## 2. Convolution of Analytic Functions

Our first result shows that the classes and are closed under convolution with prestarlike functions.

Theorem 2.1.

Let be a fixed integer and a fixed function in . Let be a convex univalent function satisfying , , and .

- (1)

Remark 2.2.

When , various known results are easily obtained as special cases of Theorem 2.1. For instance, [1, Theorem , page 336] is easily deduced from Theorem 2.1(1), while [1, Corollary , page 336] follows from Theorem 2.1(2). If is defined by (1.8), then [3, Theorem , page 110] follows from Theorem 2.1(1), and [3, Corollary , page 111] follows from Theorem 2.1(2).

Corollary 2.3.

If , then . Similarly, if , then .

Proof.

so that . By Theorem 2.1(1), it follows that .

The second result is proved in a similar manner.

Remark 2.4.

If is defined by (1.8), then Corollary 2.3 reduces to [2, Theorem , page 324].

Theorem 2.5.

Let be a fixed integer and a fixed function in . Let be a convex univalent function satisfying , and .

(1)If with respect to , then with respect to .

(2)If with respect to , then with respect to .

Since , it is evident from (2.6) that .

Remark 2.6.

Again when , known results are easily obtained as special cases of Theorem 2.5. For instance, [1, Theorem , page 337] follows from Theorem 2.5(1), and [1, Theorem , page 339] is a special case of Theorem 2.5(2).

Corollary 2.7.

Let be a fixed integer and a fixed function in . Let be a convex univalent function satisfying , . Let be the Bernardi-Libera integral transform of defined by (2.11). If , then .

The proof is similar to the proof of Corollary 2.3 and is therefore omitted.

Remark 2.8.

If is defined by (1.8), then Corollary 2.7 reduces to [2, Theorem , page 326].

## 3. Convolution of Meromorphic Functions

is a particular case of with and .

Here four classes , , , and of meromorphic functions are introduced and the convolution properties of these new subclasses are investigated. As before, it is assumed that is a fixed integer, is a fixed function in , and is a convex univalent function with positive real part in satisfying .

Definition 3.1.

The class consists of for which .

then the class coincides with investigated in [6].

Definition 3.2.

for some with and . The class consists of for which .

If , then coincides with . If is defined by (3.8), then reduces to investigated in [5]. If is defined by (3.9), then the class is the class studied in [6].

We shall require the theorem below which is a simple modification of Theorem 1.3.

Theorem 3.3.

Theorem 3.4.

Assume that is a fixed integer and is a fixed function in . Let be a convex univalent function satisfying , , and with .

Inequality (3.18) shows that .

( ) When , various known results are easily obtained as special cases of Theorem 3.4. For instance, if is defined by (3.7), then [9, Theorem , page 1265] follows from Theorem 3.4(1).

Corollary 3.6.

If , then . Similarly, if , then .

Proof.

so that . By Theorem 3.4, it follows that .

The second result is established analogously.

Remark 3.7.

Again we take note of how our results extend various earlier works. If is defined by (3.7), then [7, Proposition , page 512] follows from Corollary 3.6. If is defined by (3.8), then Corollary 3.6 yields [5, Theorem , page 4]. If is defined by (3.9), then Corollary 3.6 reduces to [6, Theorem , page 11].

Theorem 3.8.

Assume that is a fixed integer and is a fixed function in . Let be a convex univalent function satisfying , , and with .

(1)If with respect to , then with respect to .

(2)If with respect to , then with respect to .

Inequality (3.18) shows that .

( ) The function is in if and only if is in and from the first part above, it follows that . Hence .

Corollary 3.9.

Assume that is a fixed integer and is a fixed function in . Let be a convex univalent function satisfying , . Let be defined by (3.23). If , then .

The proof is analogous to Corollary 2.3 and is omitted.

Remark 3.10.

If is defined by (3.8), then Corollary 3.9 yields [5, Theorem , page 9]. If is defined by (3.9), then Corollary 3.9 reduces to [6, Theorem , page 14].

## Declarations

### Acknowledgment

The work presented here was supported in part by research grants from Universiti Sains Malaysia and University of Delhi.

## Authors’ Affiliations

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