Extremal domains associated with an analytic function I

Abstract

In this paper we investigate the following two extremal problems: A) Let F be a continuum in the extended complex plane that does not divide and let f(z) be a function analytic on F By D we denote domains in such that f(z) has a single-valued analytic continuation in D. Does there exist a domain D 0 with minimal condenser capacity B) Let f(z) be a function analytic in a neighborhood of infinity. By D we denote domains in , such that f{z) has a single-valued analytic continuation in D. Does there exist a domain D 0 with minimal logarithmic capacity It is proved that there exist extremal domains D 0 in both problems. In a second part of the paper it will be shown that these domains are uniquely determined up to a boundary set of capacity zero.