Multiply universal holomorphic functions (Q1356797)

The problem of the existence of so-called ``universal functions'' (compare W. Luh, Holomorphic monsters. [J. Approximation Theory 53, No. 2, 128-144 (1988; Zbl 0669.30020)] for the notations and the history of the topic) is generalized. The main result is the following: Let \({\mathcal O} \subset\mathbb{C}\), \({\mathcal O} \neq\mathbb{C}\), be an open set with simply connected components. Then there exists a function \(\varphi\), which is holomorphic exactly on \({\mathcal O}\) and has six universal properties at the same time. The proof uses essentially the theorems of Runge and Mergelyan.