On Schwarz-Christoffel Mappings

TL;DR

This paper advances the understanding of Schwarz-Christoffel mappings by analyzing their relationship with Blaschke products, resolving an open question, and establishing conditions for injectivity based on exterior angles.

Contribution

It extends previous work on Schwarz-Christoffel mappings, clarifies the connection with Blaschke products, and provides new criteria for injectivity and geometric properties.

Findings

Resolved an open question relating Blaschke product degrees to polygon vertices.

Established a sharp sufficient condition for injectivity based on exterior angles.

Analyzed the geometric relationship between zeros of Blaschke products and pre-vertices.

Abstract

We extend previous work on Schwarz-Chrsitoffel mappings, including the special cases when the image is a convex polygon or its complement. We center our analysis on the relationship between the pre-Schwrazian of such mappings and Blaschke products. For arbitrary Schwarz-Christoffel mappings, we resolve an open question in \cite{ChDO2} that relates the degrees of the associated Blaschke products with the number of convex and concave vertices of the polygon. In addition, we obtain a sharp sufficient condition in terms of the exterior angles for the injectivity of a mapping given by the Schwarz-Christoffel formula, and study the geometric interplay between the location of the zeros of the Blaschke products and the separation of the pre-vertices.