Bell representations of finitely connected planar domains

Abstract

In this paper, we solve a conjecture of S. Bell (1992) affirmatively. Actually, we prove that every non-degenerate n-connected planar domain Ω, where n > 1 is representable as Ω = {|f| < 1} with a suitable rational function f of degree n. This result is considered as a natural generalization of the classical Riemann mapping theorem for simply connected planar domains.