Geometry

Volume 2013 (2013), Article ID 671826, 4 pages

http://dx.doi.org/10.1155/2013/671826

Research Article

On a Subclass of Meromorphic Functions Defined by Hilbert Space Operator

Thomas Rosy and S. Sunil Varma

Department of Mathematics, Madras Christian College, Tambaram, Chennai, Tamil Nadu 600 059, India

Received 26 April 2013; Accepted 30 May 2013

Academic Editor: JinLin Liu

Copyright © 2013 Thomas Rosy and S. Sunil Varma. 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.

Abstract.

In this paper, we define a new operator on the class of meromorphic functions and define a subclass using Hilbert space operator. Coefficient estimate, distortion bounds, extreme points, radii of starlikeness, and convexity are obtained.

1. Introduction

Let denote the class of meromorphic functions

defined on . For given by (1) and given by

the Hadamard product (or convolution) [1] of and is defined by

Many subclasses of meromorphic functions have been defined and studied in the past. In particular, the subclasses ,   and   [2], and and   [3], are considered by researchers. Let be a complex Hilbert space and denote the algebra of all bounded linear operators on . For a complex-valued function analytic in a domain of the complex plane containing the spectrum of the bounded linear operator , let denote the operator on defined by the Riesz-Dunford integral [4]

where is the identity operator on and is a positively oriented simple closed rectifiable closed contour containing the spectrum in the interior domain [5]. The operator can also be defined by the following series:

which converges in the norm topology. In this paper, we introduce a subclass of defined using Hilbert space operator and prove a necessary and sufficient condition for the function to belong to this class, the distortion theorem, radius of starlikeness, and convexity. In [6], Atshan and Buti had defined an operator acting on analytic functions in terms of a definite integral. We modify their operator for meromorphic functions as follows.

Lemma 1. For given by (1), , and , if the operator is defined by

then

where .

Denote by the class of all functions with .

Definition 2. For , , a function is in the class if

for all operators with and ,   being the zero operator on .

2. Coefficient Bounds

Theorem 3. A meromorphic function given by (1) is in the class if and only if

The result is sharp for .

Proof. Let . Assume that (9) holds. Then,

and hence .

Conversely, let

This implies that

Choose ,  . Then, . As , we obtain (9).

Corollary 4. If of the form (1) is in , then

The result is sharp for the function .

Theorem 5. Let and ,  ,  ,  . Then, is in the class if and only if it can be expressed in the form , where and .

Proof. Suppose that ; then, we have

By Theorem 3, . Conversely, assume that is in the class ; then, by Corollary 4,

Set , and . Then, .

3. Distortion Bounds

In this section, we obtain growth and distortion bounds for the class .

Theorem 6. If , then

The result is sharp for .

Proof. By Theorem 3,

Therefore,

Also, if , then

Since , the above inequality becomes

Using (18), we get the result.

Theorem 7. If , then

The result is sharp for .

4. Radii Results

We now find the radius of meromorphically starlikeness and convexity for functions in the class .

Theorem 8. Let . Then, is meromorphically starlike of order ,   in , where

Proof. Let . Since is meromorphically starlike of order ,

Substituting for , the above inequality becomes

By Theorem 3,

Thus, (24) will be true if

that is,

Theorem 9. Let . Then, is meromorphically convex of order in , where

Theorem 10. The class is closed under convex combination.

Proof. Let and . Then, by Theorem 3,

Define . Then, . Now,

Thus, .

5. Hadamard Product

Theorem 11. The Hadamard product of two functions and in belongs to the class , where  .

Proof. It suffices to prove that

where

Since

By Cauchy-Schwartz inequality,

We have to find the largest such that

that is,

It is enough to find the largest such that

which yields

It follows from (38) that

Let

Clearly is an increasing function of   . Letting , we prove the assertion.

References

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