Convex Hulls and Extreme Points of Some Families of Univalent Functions

Abstract

The closed convex hull and extreme points are obtained for the functions which are convex, starlike, and close-to-convex and in addition are real on (-1,1).We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on (-1,1).Integral representations are given for the hulls of these families in terms of probability measures on suitable sets.We also obtain such a representation for the functions fiz) analytic in the unit disk, normalized and satisfying Re/'(0 > a for a < 1.These results are used to solve extremal problems.For example, the upper bounds are determined for the coefficients of a function subordinate to some function satisfying Re/'OO > a.Introduction.We shall determine the closed convex hulls and extreme points of some families of univalent functions.We utilize these results to solve specific extremal problems over certain of the families.Let A denote the unit disk [z: \z\ < 1} and let A denote the set of functions analytic in A. Then A is known to be a locally convex linear topological space where the topology is given by uniform convergence on compact subsets of A [14, p. 150].We let 5 denote the subset of A consisting of the functions / that are univalent in A and normalized so as to satisfy /(0) = 0 and/'(0) = 1.Let K, St and C denote the well-known subfamilies of S which are respectively convex, starlike and close-to-convex.We will consider the subfamilies KR, StR and CR where for any class of functions D we let DR = {/: / E 3 and /is real on (-1, 1)}.We also consider the family of functions denoted by FR which are convex in the direction of the imaginary axis and real on (-1,1).The functions in FR were studied by M. S. Robertson in [10].Further we let P(a) denote the subfamily of S consisting of those functions satisfying Re/'(z) > a where 0 < a < 1.The study of the convex hulls and extreme points of various families of univalent functions was initiated by L. Brickman, T. H. MacGregor, and D. R. Wilken in [2].It was continued by the above authors and the present author in [1].We shall use some of the basic results

Locations

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