Convex Hulls and Extreme Points of Some Families of Univalent Functions

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 $f(z)$ analytic in the unit disk, normalized and satisfying Re f’(z) > α for α < 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 f’(z) > α.