A Normality Criterion Corresponding to the Defect Relations

Abstract

Let F be a family of meromorphic functions on a domain D. We present a quite general sufficient condition for F to be a normal family. This criterion contains many known results as special cases. The overall idea is that certain comparatively weak conditions on F by local arguments lead to somewhat stronger conditions, which in turn lead to even stronger conditions on the limit function g in the famous Zalcman Lemma. Ultimately, the defect relations for g force normality of F.