Failure of weak holomorphic averaging on multiply connected domains

Abstract. We show that given a multiply-connected domain Ω with a holomorphic, multiple-valued function F whose values are bounded on compact subsets of Ω, one cannot always find a single-valued holomorphic function that “averages ” F on some neighborhood of the boundary of Ω. We then use this result to consider the extension to multiply-connected domains of two theorems (one of Alexander and Wermer, and the other of Berndtsson and Ransford) which are known to hold on the disk. In both cases we produce counterexamples showing that the extensions fail. In translating theorems about H ∞ functions on the upper half plane H into theorems about H ∞ functions on an annulus A, we have a very useful theorem. Theorem 0. If F: A → C is a multiple-valued holomorphic function whose collec-tive values {F (z)} are bounded on compact subsets of A, then there is a single-value