Differential Inequalities, Normality and Quasi-Normality

2011/11/03 by Xiaojun Liu, Liu, Xiaojun, Shahar Nevo +3
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematics and Applications #Meromorphic and Entire Functions #math.CV

paper · pdf · doi:10.48550/arxiv.1111.0844

arxiv created 2011/11/03 · openalex publication_date 2011/11/03 · arxiv updated 2011/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if D is a domain in C, alpha>1 and c>0, then the family F of functions meromorphic in D such that |f'(z)|/(1+|f(z)|^alpha)>c for every z in D is normalin D. For alpha=1, the same assumptions imply quasi-normality but not necessarily normality.

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