Convex hulls and extreme points of families of starlike and convex mappings

Abstract:

The closed convex hull and extreme points are obtained for the starlike functions of order $\alpha$ and for the convex functions of order $\alpha$. More generally, this is determined for functions which are also k-fold 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 $\alpha$. Also, the lower bound on $\operatorname {Re} \{ f(z)/z\}$ is found for each $z\;(|z| < 1)$ where f varies over the convex functions of order $\alpha$ and $\alpha \geq 0$.