On Schwarz–Christoffel mappings

Martin Chuaqui and Christian Pommerenke

Vol. 270 (2014), No. 2, 319–334

DOI: 10.2140/pjm.2014.270.319

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.

Keywords

Schwarz–Christoffel mapping, prevertices, convex, concave, univalent mapping, Blaschke product

Mathematical Subject Classification 2000

Primary: 30C20, 30C35

Secondary: 30C45

Milestones

Received: 13 May 2013

Revised: 29 November 2013

Accepted: 10 March 2014

Published: 22 August 2014