A note on extreme points of subordination classes

Abstract:

Let $s(F)$ denote the set of functions subordinate to a univalent function $F$ in $\Delta$ in the unit disc. Let $B$ denote the set of functions $\phi (z)$ analytic in $\Delta$ satisfying $|{\phi (x)}| < 1$ and $\phi (0) = 0$. Let $D = F(\Delta )$ and $\lambda (w,\partial D)$ denote the distance between $w$ and $\partial D$ (boundary of $D$). We prove that if $\phi$ is an extreme point of $B$ then $\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty }$. As a corollary we prove that if $F \circ \phi$ is an extreme point of $s(F)$ then $\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty }$.