%0 Journal Article
%T Sufficient Conditions for Meromorphically -Valent Starlikeness and Close-to-Convexity
%A B. A. Frasin
%A Tariq Al-Hawary
%A M. Darus
%J Geometry
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/497191
%X Making use of the linear operator defined by (Frasin 2012), we introduce the class of meromorphically -valent functions in the punctured unit disk . Furthermore, we obtain some sufficient conditions for starlikeness and close-to-convexity for functions belonging to this class. Several corollaries and consequences of the main results are also considered. 1. Introduction and Definitions Let denote the class of functions of the form: which are -valent in the punctured unit disk . A function in is said to be meromorphically -valent starlike of order if and only if for some . We denote by the class of all meromorphically -valent starlike of order . Further, a function in is said to be meromorphically -valent convex of order if and only if for some . We denote by the class of all meromorphically -valent convex of order . A function belonging to is said to be meromorphically -valent close-to-convex of order if it satisfies for some . We denote by the subclass of consisting of functions which are meromorphically -valent close-to-convex of order in . Many interesting families of analytic and multivalent functions were considered by earlier authors in Geometric Functions Theory (cf. e.g., [1¨C4]). Some subclasses of when were considered by (e.g.) Miller [5], Pommerenke [6], Clunie [7], Owa et al. [8], and Royster [9]. Furthermore, several subclasses of when were studied by (amongst others) Mogra et al. [10], Uralegaddi and Ganigi [11], Cho et al. [12], Aouf [13, 14], and Uralegaddi and Somanatha [15]. For a function in , Frasin [16] introduced and studied the following differential operator: and for Note that for , we have the operator introduced and studied by Frasin and Darus [17]. It easily verified from (6) that Making use of the above operator , we now introduce a new class of meromorphically and -valent functions defined as follows. Definition 1. A function is said to be a member of the class if and only if for some and for all . Note that condition (8) implies that Clearly, we have and . In this paper, we obtain some sufficient conditions for functions belonging to the class . Several corollaries and consequences of the main results are also considered. In order to derive our main results, we have to recall the following lemmas. Lemma 2 (see [18]). Let be analytic in and such that . Then if attains its maximum value on circle at a point , we have where is a real number. Lemma 3 (see [19]). Let be a set in the complex plane and suppose that is a mapping from ¡Á to which satisfies for , and for all real such that . If the function is analytic in such that for
%U http://www.hindawi.com/journals/geometry/2013/497191/