On Certain New Subclass of Meromorphic Close-to-Convex Functions

We introduce a certain new subclass of meromorphic close-to-convex functions. Such results as inclusion relationship, coefficient inequalities, distortion, and growth theorems for this class of functions are derived. 1. Introduction Let denote the class of functions of the form which are analytic in the punctured open unit disk: Let denote the class of functions given by which are analytic and convex in and satisfy the following condition: A function is said to be in the class of meromorphic starlike functions of order if it satisfies the following inequality: Moreover, a function is said to be in the class of meromorphic close-to-convex functions if it satisfies the following condition: For two functions and analytic in , we say that the function is subordinate to in and write , if there exists a Schwarz function , analytic in with and such that . Indeed, it is well known that Furthermore, if the function is univalent in , then we have the following equivalence: Recently, Wang et al. [1] introduced and investigated the class of meromorphic close-to-convex functions which satisfy the inequality where . Kowalczyk and Le？-Bomba [2] discussed the class of analytic functions related to the starlike functions; a function which is analytic in is said to be in the class , if it is satisfies the following inequality: where . ？eker [3] discussed the class of analytic functions which satisfy the following condition: where , and . Motivated essentially by the classes , and , we introduce and study the following more generalized class of meromorphic functions. Definition 1. A function is said to be in the class if it satisfies the following inequality: where is a fixed positive integer, and is given by We observe that the inequality (12) is equivalent to Since , the class is a generalization of the class . For some recent investigations on the class of close-to-convex functions, one can find them in [4–7] and the references cited therein. In the present paper, we aim at proving that the class is a subclass of meromorphic close-to-convex functions. Furthermore, some interesting results of the class are derived. 2. Preliminary Results To prove our main results, we need the following lemmas. Lemma 2. Let , where . Then for , one has Proof. Since , we have We now let Differentiating (17) logarithmically, we obtain From (18) together with (16), we can get Thus, if , we know that Lemma 3 (see [8]). Let . Then Lemma 4 (see [9]). Suppose that . Then Lemma 5 (see [10, page 105]). If the function analytic and convex in and satisfies the condition then Lemma 6 (see [10]). If