Common Universal Meromorphic Functions for Translation and Dilation Mappings

We consider translation and dilation mappings acting on the spaces of meromorphic functions on the complex plane and the punctured complex plane, respectively. In both cases, we show that there is a dense $G_{\delta}$-subset of meromorphic functions that are common universal for certain uncountable families of these mappings. While a corresponding result for translations exists for entire functions, our result for dilations has no holomorphic counterpart. We further obtain an analogue of Ansari’s Theorem for the mappings we consider, which is used as a key tool in the proofs of our main results