Extremal quasiconformality vs bounded rational approximation

Abstract

We show that, on most of the hyperbolic simply connected domains, the weighty bounded rational approximation in a natural sup norm is possible only for a very sparse set of holomorphic functions (in contrast to the integral approximation). The obstructions are caused by the features of extremal quasiconformality.The paper is devoted to the 100th anniversary of Georgii Dmitrievich Suvorov, my first university adviser and teacher. He was an outstanding mathematician and a widely talented, extremely great human being.