Hayman's alternative and normal families of non-vanishing meromorphic functions
Let \(P\) be a differential polynomial satisfying certain technical and general conditions. The author shows that a family \(\mathcal{F}\) of non-vanishing meromorphic functions in the unit disk \(\mathbb{D}\) such that \(P[f](z)\neq 1\) for all \(z\in\mathbb{D}\) and all \(f\in\mathcal{F}\), is normal. This generalizes former results of \textit{W. Schwick} [Complex Variables, Theory Appl. 32, 51--57 (1997; Zbl 0938.30021)] and of \textit{M.-L. Fang} [Acta Math. Sin. 37, 86--90 (1994; Zbl 0796.30030)]. Furthermore, the author gives corresponding Picard type theorems generalizing Hayman's alternative.
scientific article; zbMATH DE number 2103480