On extreme points of subordination families

Abstract:

Let $F$ be the set of analytic functions in $U = \{ z:|z| < 1\}$ subordinate to a univalent function $f$. Let $D = f(U)$. For $g(z) = f(\phi (z)) \in F$, let $\lambda (\theta )$ denote the distance between $g({e^{i\theta }})$ and $\partial D$ (boundary of $D$). We obtain the following results. (1) If $f’$ is Nevanlinna then $\int _0^{2\pi } {\log \lambda (\theta )d\theta = - \infty }$ if and only if \[ \int _0^{2\pi } {\log \left ( {1 - |\phi ({e^{i\theta }})|} \right )d\theta = - \infty } .\] (2) If $g$ is an extreme point of the closed convex hull of $F$ then \[ \int _0^{2\pi } {\log \left ( {1 - |\phi ({e^{i\theta }})|} \right )d\theta = - \infty } ,\] for the case when $D$ is a Jordan domain subset to a half-plane and $f’$ is Nevanlinna.