Convex hulls of some classical families of univalent functions

Abstract:

Let S denote the functions that are analytic and univalent in the open unit disk and satisfy $f(0) = 0$ and $f’(0) = 1$. Also, let K, St, ${S_R}$, and C be the subfamilies of S 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 $f(z) = z/{(1 - xz)^2},|x| = 1$ , 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 $\operatorname {Re} J(f) = \max \;\{ \operatorname {Re} \;J(g):g \in St\}$ are Koebe functions and there are only a finite number of them.