Extreme point of the unit ball in H(U) An extreme point of a convex subset $C$ of a vector space $X$ is a point which can not be expressed in the form $\lambda a+(1-\lambda)b$, with $a,b\in C$ and $0<\lambda<1$. 
For an open subset $U$ of the complex plan, $\mathbb C$, we denote by $H(U)$ the set of all holomorhphic functions on $U$. The norm on $H(U)$ is the usual $\|.\|_\infty$ norm.
What are the extreme points of the unit ball in $H(U)$?
The unit ball in $H(U)$ is the set $\{f\in H(U): \|f\|_\infty\leq 1\}$

Partial answer: If $A={\Bbb C}\setminus U$ contains at most two points then $B_1(H(U))$ consists of constants only. So assume $|A|\geq 3$ and let $\phi:U\rightarrow {\Bbb D}$ be a conformal bijection with the unit disk. On the unit disk a set of extremal points are $e_{n,\theta}=\theta z^n$, $n\geq 0$, $|\theta|=1$. And $e_{n,\theta} \circ \phi$ are then extremal points in $B_1(H(U))$. There are however functions in $B_1$ that are not generated by these extremal points.
