MULTIPLY UNIVERSAL HOLOMORPHIC-FUNCTIONS

Let O subset of C O not equal C, be an open set with simply connected components. In Theorem 1 we prove the existence of a holomorphic funct ion phi on O, which has together with all its derivatives and all its antiderivatives six universal properties at the same time (based on th e behaviour of sequences of derivatives or antiderivatives, overconver gence-phenomena, or properties of translates). In Theorem 2 we show th at the family of all functions with these universal properties is a de nse subset of the metric space H(O) of all holomorphic functions on O, if H(O) is endowed with the usual compact-open topology. (C) 1997 Aca demic Press.