Entire functions with universal translates which are bounded on each line (Q1038586)

Let \(\mathcal{M}\) denote the family of all compact sets \(K \subset \mathbb{C}\) with connected complement, and let \(\mathcal{A}(K)\) denote the family of all functions which are continuous on \(K\) and holomorphic in the interior of \(K\). In 1929, Birkoff established his well known result about the existence of an entire function with universal translates which can be formulated as follows. There exists a universal function \(\varphi\) with the property that, for every \(K \in \mathcal{M}\) and for any \(f \in \mathcal{A}(K)\), there is a sequence \(\{z_n\}\) of complex numbers with \(z_n \to \infty\) and \(\varphi(z + z_n) \to f(z)\) uniformly on \(K\). In a previous joint paper of the author with \textit{T. Gharibyan} and \textit{W. Luh} [Arch. Math., 86, No. 2, 261--267 (2006; Zbl 1094.30032)] it was shown that there exist an entire function \(\varphi\) and sequences \(\{a_n\}\) and \(\{b_n\}\) in \(\mathbb{C}\) with \(a_n \to 0\) and \(b_n \to \infty\) such that, for every \(K \subset \mathcal{M}\) and every \(f \in \mathcal{A}(K)\), there is a subsequence \(\{n_k\}\) such that \(\varphi\) is bounded on any line and \(\varphi(a_{n_k}z + b_{n_k}) \to f(z)\) uniformly on \(K\). The goal of the paper under review is to generalize this result constructing entire functions \( \varphi\) such that the function and its derivatives are universal and both (function and derivatives) are bounded on lines. Moreover, it is proved that the factor sequence \(\{a_n\}\) can be removed. The proofs of the results are based on some classical approximation theorems due to Mergelian and Arakelian. The last section of the paper is dedicated to the fact that a certain family of universal entire functions is a dense (but not residual) subset of \(H(\mathbb{C})\) endowed with the compact-open topology.