Open Access
May 2012 A generalization of Gu’s normality criterion
Xiaoyi Liu, Jianming Chang
Proc. Japan Acad. Ser. A Math. Sci. 88(5): 67-69 (May 2012). DOI: 10.3792/pjaa.88.67


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.


Download Citation

Xiaoyi Liu. Jianming Chang. "A generalization of Gu’s normality criterion." Proc. Japan Acad. Ser. A Math. Sci. 88 (5) 67 - 69, May 2012.


Published: May 2012
First available in Project Euclid: 7 May 2012

zbMATH: 1259.30027
MathSciNet: MR2925284
Digital Object Identifier: 10.3792/pjaa.88.67

Primary: 30D45

Keywords: exceptional functions , meromorphic functions , normality

Rights: Copyright © 2012 The Japan Academy

Vol.88 • No. 5 • May 2012
Back to Top