The Bloch constant of bounded analytic functions on a multiply connected domain

Abstract

Abstract.Let F be an analytic function on the bounded domain R, a multiply-connected region of the complex plane. The Bloch constant of F is defined bywhere p is a conformai universal cover of R with domain A, the open unit disk. If F is bounded, then ßF < H-FH^, the sup-norm of F . In this paper we characterize those functions F for which ßf = \\f\\oc in terms of the zeros of F when the boundary of R is the union of finitely many curves. We conclude this paper by showing the existence of extremal functions, and generalizing the results to bounded harmonic mappings on these domains.