Proceedings of the Japan Academy, Series A, Mathematical Sciences

A generalization of Gu’s normality criterion

Xiaoyi Liu and Jianming Chang

Full-text: Open access

Abstract

Let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$, $k\in\mathbf{N}$ and $\mathcal{H}$ be a normal family of meromorphic functions on $D$ such that 0 is not in $\mathcal{H}$ and $\mathcal{H}$ has no sequence that converges to 0 or $\infty$ spherically locally uniformly on $D$. If for every $f\in\mathcal{F}$, $f(z)\neq 0$, and there exists an $h_{f}\in \mathcal{H}$ such that $f^{(k)}(z)\neq h_{f}(z)$ at every $z\in D$, then the family $\mathcal{F}$ is normal on $D$. This generalizes Gu’s well-known normality criterion. It is interesting that the condition $f(z)\neq 0$ cannot be replaced by that all zeros of $f$ have large multiplicities, at least $k+3$ for instance.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 5 (2012), 67-69.

Dates
First available in Project Euclid: 7 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.pja/1336394839

Digital Object Identifier
doi:10.3792/pjaa.88.67

Mathematical Reviews number (MathSciNet)
MR2925284

Zentralblatt MATH identifier
1259.30027

Subjects
Primary: 30D45: Bloch functions, normal functions, normal families

Keywords
Meromorphic functions normality exceptional functions

Citation

Liu, Xiaoyi; Chang, Jianming. A generalization of Gu’s normality criterion. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 5, 67--69. doi:10.3792/pjaa.88.67. https://projecteuclid.org/euclid.pja/1336394839


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