Degree and holomorphic extendibility

Recently it was shown that if D is a bounded domain in ℂ whose boundary consists of a finite number of pairwise disjoint simple closed curves, then a continuous function f on bD extends holomorphically through D if and only if, for each g ∈ A(D) such that f + g has no zero on bD, the degree of f + g is non-negative (which, for these special domains, is equivalent to the fact that the change of argument of f + g along bD is non-negative). Here A(D) is the algebra of all continuous functions on D which are holomorphic on D. This fails to hold for general domains, and generalizing to more general domains presents a major problem that often requires a much larger class of functions g. It is shown that the preceding theorem still holds in the case when D is a bounded domain in ℂ such that D is finitely connected and such that D is equal to the interior of D.

Single-valued conjugates and holomorphic extendibility

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.