On Schwarz-Christoffel mappings

Abstract

We extend previous work on Schwarz-Christoffel 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-Schwarzian of such mappings and Blaschke products. For arbitrary Schwarz-Christoffel mappings, we resolve an open question from earlier work of Chuaqui, Duren and Osgood 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 prevertices. © 2014 Mathematical Sciences Publishers.