Normal families of holomorphic functions and shared values (Q1601470)

Starting from Picard type theorems many normality criteria have been proved [cf. \textit{J. L. Schiff}, Normal Families (1993; Zbl 0770.30002)]. \textit{W. Schwick} [Arch. Math. 59, No. 1, 50-54 (1992; Zbl 0758.30028)] initiated the problem of finding out normality criteria starting from the uniqueness theorems of an entire or a meromorphic function and its derivative. In the paper the authors prove three normality criteria for families of holomorphic functions in a domain based on the uniqueness theorems. Two of the theorems improve an earlier result. They give examples to show that these two theorems cannot be extended to a family of meromorphic functions. The main tool of the paper is an amalgamation of the Nevanlinna theory and a Zalcman type lemma.