Extreme points and support points of conformal mappings

Abstract

There are three types of results in this paper. The first, extending a representation theorem on a conformal mapping that omits two values of equal modulus. This was due to Brickman and Wilken. They constructed a representation as a convex combination with two terms. Our representation constructs convex combinations with unlimited number of terms. In the limit one can think of it as an integration over a probability space with the uniform distribution. The second result determines the sign of $\Re L(\overline{z}_0(f(z))^2)$ up to a remainder term which is expressed using a certain integral that involves the L\"owner chain induced by $f(z)$, for a support point $f(z)$ which maximizes $\Re L$. Here $L$ is a continuous linear functional on $H(U)$, the topological vector space of the holomorphic functions in the unit disk $U=\{z\in\mathbb{C}\,|\,|z|<1\}$. Such a support point is known to be a slit mapping and $f(z_0)$ is the tip of the slit $\mathbb{C}-f(U)$. The third demonstrates some properties of support points of the subspace $S_n$ of $S$. $S_n$ contains all the polynomials in $S$ of degree $n$ or less. For instance such a support point $p(z)$ has a zero of its derivative $p'(z)$ on $\partial U$.