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 possess “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 paper establishes a strong unboundedness property for such functions near every boundary point. The result is new even in the case of the disc, where it strengthens work of several authors.
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