The Extreme and Support Points of a New Class of Analytic Functions with Positive Real Part

Suppose that . Let denote the set of functions that are analytic in and satisfy and . In this paper, we investigate the extreme points and support points of . 1. Introduction By , we denote the space of functions analytic in the unit disk . Endowed with the topology of uniform convergence on compact subsets of the unit disk, is a locally convex topological vector space. Let be a topological vector space and a subset of . If , then is called an extremal subset of provided that whenever , where , and , then and both belong to . An extremal subset of consisting of just one point is called an extreme point of . Thus, an element is an extreme point of if and only if is not a proper convex combination of any two distinct points in . The set of all extreme points of is denoted by . It is apparent that if is an extremal subset of , then . If is a locally convex topological vector space and is a nonempty compact subset of , then is nonempty [1, page 44], [2, page 181]. For any subset of , we use to denote the closed convex hull of . If is a compact subset of the locally convex topological vector space , then, by Krein-Milman theorem [1, page 44], [2, page 182], . Let be the set of all functions which are analytic, have positive real part in , and satisfy . Then is a compact subset of [1, page 39]. It is well known that [1, page 48], [3–5]. Bellamy and Tkaczyńska [6] investigated the extreme points of some classes of analytic functions with positive real part and a prescribed set of coefficients. Peng [7] investigated the extreme points of a class of analytic functions with positive real part and a prescribed set of values. Suppose that is a compact subset of . A function is called a support point of if and there is a continuous linear functional on such that is nonconstant on and The set of all support points of is denoted by . Hallenbeck and MacGregor [1, page 94], [8] proved that the set consists of all functions which may be written as where , , and ？？ . The author [9] investigated the support points of a class of analytic functions with positive real part and a prescribed set of coefficients. Suppose that . Let denote the set of functions that are analytic in and satisfy and . It is apparent that if and only if with some and . Thus, it is easy to prove that is a compact subset of . In this paper we investigate the extreme points and support points of . In some ways, the results we obtained generalize the results of Holland, Hallenbeck, and MacGregor. 2. Main Results Theorem 1. is an extreme point of if and only if where . Proof. Suppose that where ,？？ ,？？ ,