Certain New Classes of Analytic Functions with Varying Arguments

We introduce certain new classes and , which represent the κ uniformly starlike functions of order α and type β with varying arguments and the κ uniformly convex functions of order α and type β with varying arguments, respectively. Moreover, we give coefficients estimates, distortion theorems, and extreme points of these classes. 1. Introduction Let denote the class of functions of the following form: that are analytic and univalent in the open unit disc . Definition 1 (see [1]). Let denote the subclass of consisting of functions of the form (1) and satisfy the following inequality: Also let denote the subclass of consisting of functions of the form (1) and satisfy the following inequality: It follows from (2) and (3) that The class denote the class of uniformly starlike functions of order and type and the class denotes the class of uniformly convex functions of order and type . Specializing parameters ,？？ , and , we obtain the following subclasses studied by various authors:(i) and (see [2, 3]);(ii) and (see [4]);(iii) and (see [5, 6]);(iv) and (see [4, 7–10]). Also we note that which are the uniformly starlike functions of order and type and uniformly convex functions of order and type , respectively. Definition 2 (see [11]). A function of the form (1) is said to be in the class if and for all . If furthermore there exist a real number such that , then is said to be in the class . The union of taken over all possible sequences and all possible real numbers is denoted by . Let denote the subclass of consisting of functions . Also Let denote the subclass of consisting of functions . In this paper we obtain coefficient bounds for functions in the classes and , respectively, further we obtain distortion bounds and the extreme points for functions in these classes. 2. Coefficient Estimates Unless otherwise mentioned, we assume in the reminder of this paper that , and . We shall need the following lemmas. Lemma 3. The sufficient condition for given by (1) to be in the class is that Proof. It suffices to show that inequality (2) holds true. Upon using the fact that then inequality (2) may be written as or where and , then condition (2) or (9) is equivalent to We note that Using (11) and (12), then we can obtain the following inequality: The expression is bounded below by if or Hence the proof of Lemma 3 is completed. By using (4) and (6) we can obtain the following lemma. Lemma 4. A function of the form (1) is in the class if Remark 5. Putting in Lemmas 3 and 4, we obtain the results obtained by Shams et al. [3, Theorems 2.1, 2.2, resp.]. In the following