On extreme points of families described by subordination

Abstract:

Let $s\left ( F \right )$ denote the set of analytic functions in $D = \{ z:|z| < 1\}$ subordinate to an analytic function $F$. It is shown that if $F$ is a polynomial then the extreme points of the closed convex hull of $s(F) \subset \{ F \circ \phi :\phi \in {\text {extreme}}\;{\text {points}}\;{\text {of}}\;B(H_0^\infty )\}$. Also if $F(z) = {((z - \alpha )/(1 - \bar \alpha z))^n},|\alpha | < 1$ and $n$ is a positive integer then the extreme points of the closed convex hull of $s(F) = \{ F \circ \phi :\phi \in {\text {extreme}}\;{\text {points}}\;{\text {of}}\;B(H_0^\infty )\}$. An analogue of Ryff’s theorem, and other results related to subordination in Bergman spaces have been obtained.