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)( - 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)( - 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 functionsf(z)f(z)analytic in the unit disk, normalized and satisfyingRe⁡f′(z)>α\operatorname {Re} f’(z) > \alphaforα>1\alpha > 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 satisfyingRe⁡f′(z)>α\operatorname {Re} f’(z) > \alpha.