Support points of subordination families

Abstract:

Let $s(F)$ denote the set of functions subordinate to a function $F$ analytic in the unit disc $\Delta$. Let $Hs(F)$ denote the closed convex hull of $s(F)$ and supp $s(F)$ the set of support points of $s(F)$. We prove the following Theorem. Let $F$ be analytic in $\Delta$ and satisfy (1) $Hs(F) = \{ \int _{\partial \Delta } {F(xz)d\mu (x):\mu \;{\text {a}}\;{\text {probablity}}\;{\text {measure}}\;{\text {on}}\;\partial \Delta } \}$ and (2) $F(z) = G(z)/{(z - {x_0})^\alpha }$ where $G$ is analytic in $\Delta$, continuous in $\bar \Delta$, $G({x_0}) \ne 0$ and $\alpha > 1$. Then supp $s(F) = \left \{ {F(xz):|x| = 1} \right \}$.