Single-valued conjugates and holomorphic extendibility

Abstract

Let D be a bounded domain in the complex plane whose boundary consists of m ≥ 2 pairwise disjoint simple closed curves and let A(bD) be the algebra of all continuous functions on bD which extend holomorphically through D. We show that a continuous function Φ on bD belongs to A(bD) if for each g ∈ A (bD) the harmonic extension of Re(gΦ) to D has a single-valued conjugate.