International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 302583 | 11 pages | https://doi.org/10.1155/2010/302583

Classes of Meromorphic Functions Defined by the Hadamard Product

Academic Editor: Vladimir Mityushev
Received02 Jul 2009
Revised21 Nov 2009
Accepted05 Jan 2010
Published02 Feb 2010

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 [510].

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 [510].

References

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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.


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