Research Article | Open Access

# Classes of Meromorphic Functions Defined by the Hadamard Product

**Academic Editor:**Vladimir Mityushev

#### Abstract

The object of the present paper is to introduce new classes of meromorphic functions with varying argument of coefficients defined by means of the Hadamard product (or convolution). Several properties like the coefficients bounds, growth and distortion theorems, radii of starlikeness and convexity, and partial sums are investigated. Some consequences of the main results for well-known classes of meromorphic functions are also pointed out.

#### 1. Introduction

Let denote the class of functions which are *analytic* in , where

with a simple pole in the point By we denote the class of functions of the form

Also, by we denote the class of functions of the form (1.2) for which

For we obtain the classes and of functions with positive coefficients and negative coefficients, respectively.

Motivated by Silverman [1], we define the class

It is called the class of functions with varying argument of coefficients.

Let A function is said to be *meromorphically convex of order * in if

A function is said to be *meromorphically starlike of order * in if

We denote by the class of all functions , which are meromorphically convex of order in and by we denote the class of all functions which are meromorphically starlike of order in . We also set

It is easy to show that for a function the condition (1.6) is equivalent to the following:

Let be a subclass of the class . We define *the radius of starlikeness of order * and the radius of convexity of order for the class by

respectively.

Let functions be analytic in We say that the function is *subordinate* to the function , and write (or simply ) if there exists a function analytic in , such that

In particular, if is univalent in , we have the following equivalence:

For functions of the form

by we denote *the Hadamard product *(or* convolution*) of and , defined by

Let be real parameters, and let be given functions from the class

By we denote the class of functions such that and

Moreover, let us define

where the functions have the form

and the sequences are nonnegative real, with

Moreover, let us put

It is easy to show that

The object of the present paper is to investigate the coefficient estimates, distortion properties and the radii of starlikeness and convexity, and partial sums for the classes of meromorphic functions with varying argument of coefficients. Some remarks depicting consequences of the main results are also mentioned.

#### 2. Coefficients Estimates

First we mention a sufficient condition for functions to belong to the class .

Theorem 2.1. *Let be defined by (1.18), . If a function of the form (1.2) satisfies the condition
**
then belongs to the class .*

*Proof. *A function of the form (1.2) belongs to the class if and only if there exists a function such that
or equivalently
Thus, it is sufficient to prove that
Indeed, letting we have
whence

Theorem 2.2. *Let be a function of the form (1.2), with (1.3). Then belongs to the class if and only if the condition (2.1) holds true.*

*Proof. *In view of Theorem 2.1, we need only to show that each function from the class satisfies the coefficient inequality (2.1). Let Then by (2.3) and (1.2), we have
Therefore, putting and applying (1.3), we obtain
It is clear, that the denominator of the left hand said cannot vanish for Moreover, it is positive for and in consequence for Thus, by (2.7), we have
which, upon letting , readily yields the assertion (2.1).

From Theorem 2.2, we obtain coefficients estimates for the class .

Corollary 2.3. *If a function of the form (1.2) belongs to the class then
**
where is defined by (1.18). The result is sharp. The functions of the form
**
are the extremal functions. *

#### 3. Distortion Theorems

From Theorem 2.2, we have the following lemma.

Lemma 3.1. *Let a function of the form (1.2) belong to the class If the sequence defined by (1.18) satisfies the inequality
**
then
**
Moreover, if
**
then
*

Theorem 3.2. *Let a function belong to the class If the sequence defined by (1.18) satisfies (3.1), then
**
Moreover, if (3.3) holds, then
**
The result is sharp, with the extremal function of the form (2.10).*

*Proof. *Let a function of the form (1.2) belong to the class Since
then by Lemma 3.1 we have (3.5). Analogously we prove (3.6).

#### 4. The Radii of Convexity and Starlikeness

Theorem 4.1. *The radius of starlikeness of order for the class is given by
**
where is defined by (1.18).*

*Proof. *The function of the form (1.2) is meromorphically starlike of order in the disk if and only if it satisfies the condition (1.8). Since
putting the condition (1.8) is true if
By Theorem 2.2, we have
Thus, the condition (4.3) is true if
that is, if
It follows that each function is meromorphically starlike of order in the disk , where is defined by (4.1). Moreover, the radius of starlikeness of the functions defined by (2.10) is given by
Thus we have (4.1).

Theorem 4.2. *The radius of convexity of order for the class is given by
**
where is defined by (1.18).*

*Proof. *The proof is analogous to that of Theorem 4.1, and we omit the details.

#### 5. Partial Sums

Let be a function of the form (1.2). Motivated by Silverman [2] and Silvia [3] (see also [4]), we define the partial sums defined by

In this section, we consider partial sums of functions from the class and obtain sharp lower bounds for the real part of ratios of to and to

Theorem 5.1. *Let and let the sequence defined by (1.18), satisfy the inequalities
**
If a function belongs to the class then
**
The bounds are sharp, with the extremal function of the form (2.10).*

*Proof. *Let a function of the form (1.2) belong to the class Then by (5.2) and Theorem 2.2, we have
If we put
then it suffices to show that
or
Applying (5.5), we find that
which readily yields the assertion (5.3). In order to see that gives the result sharp, we observe that for we have
Similarly, if we take
and make use of (5.5), we can deduce that
which leads us immediately to the assertion (5.4). The bound in (5.4) is sharp for each with the extremal function given by (2.10).

Theorem 5.2. *Let and let the sequence defined by (1.18), satisfy the inequalities (5.2). If a function belongs to the class then
**
The bounds are sharp, with the extremal function of the form (2.10).*

*Proof. *The proof is analogous to that of Theorem 5.1, and we omit the details.

*Remark 5.3. *We observe that the obtained results are true if we replace the class by

#### 6. Concluding Remarks

We conclude this paper by observing that, in view of the subordination relation (1.18), by choosing the functions and , we can define new classes of functions. In particular, the class

contains functions such that

A function belongs to the class

if it satisfies the condition

The class is related to the class of starlike function of order . In particular, we have the following relationships:

Let be a convex parameter. A function belongs to the class

if it satisfies the condition

The classes and generalize well-known important classes, which were investigated in earlier works; see for example [5–10].

If we apply the results presented in this paper to the classes discussed above, we can obtain several additional results. Some of these results were obtained in earlier works; see for example [5–10].

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#### Copyright

Copyright © 2010 J. Dziok. 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.