Meromorphic functions sharing three values with a derivative
The following result is proved: Theorem. Let \(f\) be a meromorphic and nonconstant function and \(f^{(k)}\) its \(k\)-th derivative (\(k\in\mathbb{N}\) is a fixed number). If \(f\) and \(f^{(k)}\) do not take three (pairwise distinct) complex values then \(f\equiv f^{(k)}\). This theorem generalizes the analogous proposition of \textit{E. Mues} and \textit{N. Steinmetz} [Manuscr. Math. 29, 195-206 (1979; Zbl 0416.30028)] devoted to the case \(k=1\). The result is obtained due to traditional technics used in the value distribution theory of meromorphic functions (Nevanlinna theory).
scientific article; zbMATH DE number 147377