Convex hulls and extreme points of families of starlike and convex mappings

The closed convex hull and extreme points are obtained for the starlike functions of order a and for the convex functions of order a. More generally, this is determined for functions which are also Mold symmetric. Integral representations are given for the hulls of these and other families in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to or majorized by some function which is starlike of order a. Also, the lower bound on Re (/ (z) /z} is found for each z (\ 0. Introduction. In this paper we determine the closed convex hulls and extreme points of families of functions which are generalizations of the starlike and convex mappings. These results allow us to solve a number of extremal problems over related families of analytic functions. Let A denote the unit disk {z G C: \ < 1) and let A denote the set of functions analytic in A. Then A is a locally convex linear topological space with respect to the topology given by uniform convergence on compact subsets of A. Let S be the subset of A consisting of the functions/ that are univalent in A and satisfy / (0) = 0 and f' (0) = 1. Let K and Si denote the subfamilies of S of convex and starlike mappings; that is, / e K if / (A) is convex, and/ e St if/ (A) is starlike with respect to 0. The problem of studying the convex hulls and the extreme points of various families of univalent functions was initiated by three of the authors in 2. We shall take advantage of some of the basic results obtained there and generally use the same notation with the exception that !F shall now denote the closed convex hull of a family of functions F. @F shall denote the set of extreme points of F. Theorems 2 and 3 in 2 completely determined the sets K, < #, S/ and 5/. The present paper contains generalizations of these results. We consider the family, denoted St (a), of starlike functions of order a introduced in 11 by M. S. Robertson. A function/analytic in A belongs to St (a)