Concave conformal mappings and pre-vertices of Schwarz-Christoffel mappings (Q2845866)

A concave mapping is a conformal, meromorphic function that maps the unit disc onto a complement of a convex, compact set. The paper under review considers Schwarz-Christoffel mappings that are concave mappings onto complements of the closed bounded sets whose boundaries are convex polygons. Specifically, the paper undertakes to locate the points on the unit circle that are mapped onto the vertices of the polygon in question. These points are called pre-vertices. It is shown that a set S on the unit circle is the set of pre-vertices of a convex polygon under a concave Schwarz-Christoffel mapping with a pole at the origin if and only if the origin is in the interior of the convex hull of \(S\). The latter is also equivalent to the existence of a finite Blaschke product \(B\) of specified degree such that the points of S are solutions of the equation \(z^2 B(z) = 1\). Finally, the author's study of Schwarz-Christoffel mappings leads to an insight that will be interesting to anyone working with finite Blaschke products.