Universality of holomorphic functions bounded on closed sets
Let \(G\) be a domain of the complex plane \(\mathbb C\) and let \((G, \star)\) be a topological group such that \(G\) carries the euclidean topology and for each \(a \in G\) the translation selfmapping \(\tau_a: z \in G \mapsto z \star a\) is holomorphic in \(G\). A holomorphic function \(f\) in \(G\) (\(f\in H(G)\)) is \(\tau\)-universal if and only if there exists a sequence \((a_n)\) in \(G\) such that the set \(\{f\circ \tau_{a_n}\colon n\in\mathbb N\}\) is dense in \(H(G)\). In this paper the authors solve some questions concerning the existence of \(\tau\)-universal functions when boundedness conditions on certain closed sets \(F\) are added. For it, the topology and geometry of these \(F\) play an important role. In Theorem 3.1 it is established that the Chebyshev radius \(\rho (G \setminus F)\) must be \(+ \infty\). Recall that if \(A\subseteq \mathbb C\), \(\rho(A) = \sup \{r>0:\) there exists a closed ball \(B\) of radius \(r\) with \(B \subset A\}\). If \(G=\mathbb C\), then it is shown (Theorem 3.3) that there exists a \(\tau\)-universal entire function that is bounded on \(F\subseteq\mathbb C\) if and only if there exists an Arakelyan subset \(F_0\) of \(\mathbb C\) such that \(F\subset F_0\) and \(\rho(\mathbb C\setminus F_0)=\infty\). When \(G=\) the unit disk \(\mathbb D\) and \(F\) is an Arakelian subset of \(\mathbb D\), the existence of such \(\tau\)-universal function is equivalent to \(\rho (\mathbb D \setminus F)=+ \infty \) (Theorem 3.4). A similar result (Theorem 3.6) is obtained for harmonic functions in \(\mathbb R^n\). Finally, the authors extend \textit{G. Costakis}' and \textit{M. Sambarino}'s result [Adv. Math. 182, 278--306 (2004; Zbl 1066.47005)] by showing (Theorem 4.2) that for an infinite-order differential operator \(T\), there exists a \(T\)-universal entire function \(f\) such that \(\lim_{r\to\infty} f(r\alpha)=0\) for each \(t\) in a prescribed subset of first category in \(\partial\mathbb D\).
scientific article; zbMATH DE number 5014243
scientific article