Subordination families and extreme points

Abstract:

Let $s(F)$ denote the set of functions subordinate to a univalent function $F$ in $\Delta$ the unit disk. Let ${B_0}$ denote the set of functions $\phi (z)$ analytic in $\Delta$ satisfying $|\phi (z)| < 1$ and $\phi (0) = 0$. We prove that if $f = F \circ \phi$ is an extreme point of $s(F)$, then $\phi$ is an extreme point of ${B_0}$. Let $D = F(s)$ and $\lambda (w, \partial D)$ denote the distance between $w$ and $\partial D$ (boundary of $D$). We also prove that if $\phi$ is an extreme point of ${B_0}$ and $|\phi ({e^{it}})| < 1$ for almost all $t$, then $\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}}){e^{i\theta }}), \partial D) dt = - \infty }$ for almost all $\theta$.