Multiply Universal Holomorphic Functions Original Research Article

Let O⊂C, O≠C, be an open set with simply connected components. In Theorem 1 we prove the existence of a holomorphic functionφon O, which has together with all its derivatives and all its antiderivatives six universal properties at the same time (based on the behaviour of sequences of derivatives or antiderivatives, overconvergence-phenomena, or properties of translates). In Theorem 2 we show that the family of all functions with these universal properties is a dense subset of the metric spaceH(O) of all holomorphic functions on O, ifH(O) is endowed with the usual compact-open topology.