Class of Multivalent Analytic Functions Defined by a Linear Operator

Making use of the linear operator defined in (Prajapat, 2012), we introduce the class of analytic and -valent functions in the open unit disk . Furthermore, we obtain some sufficient conditions for starlikeness and close-to-convexity and some angular properties 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 analytic and -valent in the open unit disk and . In particular, we set . A function is said to be in the class of -valently starlike of order in if and only if it satisfies the inequality Furthermore, a function is said to be in the class of -valently close-to-convex of order in if and only if it satisfies the inequality In particular, we write and , where and are the usual subclasses of consisting of functions which are starlike and close-to-convex in , respectively. In [1], Prajapat define a generalized multiplier transformation operator as follows: We see that for , we have where and . It is readily verified from (5) that We observe that the operator generalize several previously studied familiar operators, and we will show some of the interesting particular cases as follows:(i) (see [2]);(ii) (see [3, 4]);(iii) (see [5–7]);(iv) (see [8, 9]);(v) (see [10]);(vi) (see [11]);(vii) (see [12]);(viii) (see [13]).(For other generalizations of the operator , see [1]). Making use of the above operator , we introduce the class of analytic 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 (7) implies that We note that , the class which has been introduced and studied by the author in [14]. Also, we have , . The class is the class which has been introduced and studied by Frasin and Darus [15] (see also [16, 17]). In this paper, we obtain some sufficient conditions and some angular properties 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 , one has where is a real number. Lemma 3 (see [19]). Let be a set in the complex plane and suppose that is a mapping from to which satisfies for , and for all real such that . If the function is analytic in such that for all , then . Lemma 4 (see [20]). Let be analytic in with and for all .