A normality criterion corresponding to the defect relations

Abstract

Let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$. We present a quite general sufficient condition for $\mathcal{F}$ to be a normal family. This criterion contains many known results as special cases. The overall idea is that certain comparatively weak conditions on $\mathcal{F}$ locally 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 $\mathcal{F}$.

12 pages, refereed version, minor changes

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• Meromorphic functions that partially share values with their first derivative