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Abstract. Let S denote the functions that are analytic and univalent in the open unit disk and satisfy /(0) = 0 and/'(0) = 1. Also, let K, St, SR, and C be the subfamilies of 5 consisting of convex, starlike, real, and close-to-convex mappings, respectively. The closed convex hull of each of these four families is determined as well as the extreme points for each. Moreover, integral formulas are obtained for each hull in terms of the probability measures over suitable sets. The extreme points for each family are particularly simple; for example, the Koebe functions f(z) = z/(l-xz)2, \x\ = 1, are the extreme points of cl co St. These results are applied to discuss linear extremal problems over each of the four families. A typical result is the following: Let J be a "nontrivial" continuous linear functional on the functions analytic in the unit disk. The only functions in St that satisfy Re/(/) = max {ReJ(g) : geSt} are Koebe functions and there are only a finite number of them.Introduction. We shall be concerned with the closed convex hulls of various families of functions that are analytic and univalent in the open unit disk A={z eC : \z\< 1}. For each family considered we obtain integral representations for the closed convex hull, and we determine all the extreme points. In each case the extreme points are strikingly simple and familiar functions. Thus we obtain a powerful tool for solving linear extremum problems over such families.Let us establish some notation and outline our main results. We shall let A denote the set of all functions analytic in A. With the natural topology of uniform convergence on compact subsets of A, A is a locally convex linear topological space [15, p. 150]. Let S be the subset of A consisting of functions / that are univalent

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ABSTRACT. 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 (\z\ < 1) where/varies over the convex functions of order a and a > 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: \z\ < 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

Expressing &(z) and fe'(z) in terms of 0(z) gives