On the preimage sets of entire functions
\textit{F. Gross} gave in Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes 599, 51-67 (1977; Zbl 0357.30007) the following definition: A countable discrete set \(\Omega\) is defined to be a ''nontrivial preimage set'' NPS if there exists a nonlinear entire function f and a set of distinct complex values \(2\leq| s| <\infty\) (here \(| s|\) denotes the cardinality of s) such that \(f^{-1}(s)=\Omega,\) where multiplicities are counted accordingly. It should be mentioned that this definition is one aspect of the general problem discussed by \textit{R. Nevanlinna} in Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Gedächtnisfeier Karl Weierstraß, 579-582 (1966; Zbl 0144.332). In the paper under review the problem is treated to give a complete characterization of the set of entire functions f associated with an NPS \(\Omega\). This problem is some extension of the problem treated in several papers under the terminology ''functions that share values''. The author gives a description of the set of functions with the NPS \(\Omega =\{0,\pm\pi,\pm 2\pi,...\}\). Further the author gives a negative answer to the following question put up by F. Gross and C. C. Yang: For any given NPS \(\Omega\), does there exist some finite set \(\Omega_ 0\), such that \(\Omega\cup \Omega_ 0\) or \(\Omega -\Omega_ 0\) is also an NPS? (This question and the answer can be considered as extension of results given by \textit{M. Ozawa} [see for example Kodai Math. J. 2, 194-199 (1979; Zbl 0416.30026)].)
scientific article; zbMATH DE number 3873666