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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

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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  3. G. Murugusundaramoorthy and T. Rosy, “Subclasses of analytic functions associated with Fox-Wright's generalized hypergeometric functions based on Hilbert space operator,” Studia Universitatis Babeş-Bolyai, vol. 56, no. 3, pp. 61–72, 2011. View at MathSciNet
  4. N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Wiley-Interscience, New York, NY, USA, 1958. View at Zentralblatt MATH · View at MathSciNet
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