Mapping Degree and Holomorphic Extendibility on Infinitely Connected Domains

Let D ⊂ C D ⊂ C be a bounded infinitely connected domain obtained from the open unit disc by deleting a sequence of pairwise disjoint closed subdiscs with centeres on the positive real axis, accumulating only at 1. Let A(D) be the algebra of all continuous functions on D̄ D ¯ which are holomorphic on D . We prove, that a continuous function f on bD extends to a function in A ( D ) if and only if for each g ∈ A(D) g ∈ A ( D ) such that f+g = 0 f + g ≠ 0 on bD , the degree of $$f+g$$ f + g is non-negative.