On a Subclass of Meromorphic Functions Defined by Hilbert Space Operator

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