Extreme points and support points of conformal mappings

TL;DR

This paper extends representation theorems for conformal mappings, analyzes the sign of a specific real part involving support points, and explores properties of support points within polynomial subclasses of conformal maps.

Contribution

It introduces a generalized convex combination representation for conformal maps omitting two equal modulus values, and studies support points and their properties in the context of conformal and polynomial mappings.

Findings

Extended representation theorem with infinite convex combinations.

Determined the sign of a real part involving support points and L"owner chains.

Identified properties of support points in polynomial subclasses, such as zeroes of derivatives on the boundary.

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 $\partial U$.