A New Subclass of Meromorphic Close-to-Convex Functions

A new subclass ( , , ) of meromorphic close-to-convex functions, defined by means of subordination, is investigated. Some results such as inclusion relationship, coefficient inequality, convolution property, and distortion property for this class are derived. The results obtained here are extension of earlier known work. 1. Introduction Let denote the class of functions of the form which are analytic in the punctured open unit disk . For any two analytic function and in , we say that is subordinate to in , written as , if there exists a Schwarz function such that , for . In particular, if is univalent in , then is subordinate to , if and only if . If is given by (1) and is given by then the Hadamard product (or convolution) of the function and is defined by A function is said to be in the class of meromorphic starlike functions of order if it satisfies the inequality Moreover, a function is said to be in the class of meromorphic close to convex function if it satisfies the condition Recently, Wang et al. [1] introduced and discussed the class of meromorphic functions which satisfies the inequality where . The class is very close to the interesting subclass of close-to-convex function for analytic function introduced and studied by Gao and Zhou [2]. Many classes related to the class have been further studied by some authors. Especially, Wang et al. [3, 4], Kowalczyk and Le？-Bomba [5], Xu et al. [6], ？eker [7], and Cho et al. [8] introduced generalization of and obtained some properties for analytic functions in each class. More recently, by means of subordination, Sim and Kwon [9] discussed a subclass of the class . A function is said be in the class if it satisfies the following subordination relation: where and . Here, the assumption that is meromorphic starlike function of order makes the function meromorphic starlike. So, instead of in (6) and (7), we can consider , because if , then is also a meromorphic starlike function, which motivates us to define a new subclass of meromorphic close-to-convex functions as follows. Definition 1. A function is said to be in the class if there exists , such that The class provides a generalization of the classes given by Sim and Kwon [9] and Wang et al. [1]. The transformation involved in the class is analytic and convex univalent in . Moreover, In this paper, we aim at proving results such as inclusion relationship, coefficient bounds, and distortion theorem for the class . 2. Properties of Meromorphic Starlike Functions In beginning, we prove the following result of meromorphic starlike functions. Theorem 2.