Subclasses of Meromorphic Functions Associated with Convolution

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

Let be the set of all analytic functions defined in the unit disk . We denote by the class of normalized analytic functions defined in . For two functions and analytic in , the function is subordinate to , written as

(1.1)

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

A function is starlike if is subordinate to and convex if is subordinate to . Ma and Minda [1] gave a unified presentation of these classes and introduced the classes

(1.2)

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

The convolution or the Hadamard product of two analytic functions and is given by

(1.3)

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].

Let denote the class of meromorphic functions of the form

(1.4)

which are analytic and univalent in the punctured unit disk . For we recall that the classes of meromorphic starlike, meromorphic convex, meromorphic close-to-convex, meromorphic -convex (Mocanu sense), and meromorphic quasi-convex functions of order , denoted by , , , and respectively, are defined by

(1.5)

The convolution of two meromorphic functions and , where is given by (1.4) and , is given by

(1.6)

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.

Let denote the class of starlike functions of order . The class of prestarlike functions of order is defined by

(1.7)

for , and

(1.8)

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

Let , , and . Then, for any analytic function ,

(1.9)

,

where denotes the closed convex hull of .

Theorem 1.2 (see [13]).

Let be convex in and with . If is analytic in with , then

(1.10)

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.

The class consists of functions satisfying in and the subordination

(2.1)

Remark 2.2.

If , then coincides with , where

(2.2)

Definition 2.3.

The class consists of functions satisfying in and the subordination

(2.3)

Definition 2.4.

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

(2.4)

Definition 2.5.

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

(2.5)

Definition 2.6.

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

(2.6)

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.

Define the function by

(3.1)

For , it follows that

(3.2)

and therefore,

(3.3)

Hence . A computation shows that

(3.4)

Theorem 1.1 yields

(3.5)

and because , it follows that

(3.6)

Theorem 3.2.

The function if and only if

Proof.

The results follow from the equivalence relations

(3.7)

Theorem 3.3.

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

Proof.

Since , it follows that

(3.8)

and thus

(3.9)

Let

(3.10)

A similar computation as in the proof of Theorem 3.1 yields

(3.11)

Inequality (3.9) shows that . Therefore Theorem 1.1 yields

(3.12)

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 .

Let the function be defined by

(3.13)

A similar computation as in the proof of Theorem 3.1 yields

(3.14)

Since and , it follows from Theorem 1.1 that

(3.15)

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.

Define the function by

(3.16)

and the function by

(3.17)

For , it follows that Note also that

(3.18)

• (i) Since and

  

  (3.19)

Theorem 1.2 yields Hence .

• (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.

If , then there exists such that

(3.21)

Also,

(3.22)

Since , by Theorem 3.2 , . Hence .

Conversely, if , then

(3.23)

for some . Let be such that . The proof is completed by observing that

(3.24)

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.

under the conditions of Theorem 3.3.

Proof.

If , it follows from Corollary 3.12 that The subordination

(3.25)

gives . Therefore

Open Problem

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