Universal Taylor series for non-simply connected domains

Abstract

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that have “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in C \\ Ω that have connected complement. This note shows that this phenomenon can break down for non-simply connected domains Ω, even when C \\ Ω is compact. This answers a question of Melas and disproves a conjecture of Müller, Vlachou and Yavrian.