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


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

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

First available in Project Euclid: 7 May 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Meromorphic functions normality exceptional functions


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.

Export citation


  • Y. X. Gu, A normal criterion of meromorphic families, Sci. Sinica, Math. Issue (I), (1979), 267–274.
  • W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  • S. Nevo, X. Pang and L. Zalcman, Quasinormality and meromorphic functions with multiple zeros, J. Anal. Math. 101 (2007), no. 1, 1–23.
  • X. Pang, D. Yang and L. Zalcman, Normal families of meromorphic functions whose derivatives omit a function, Comput. Methods Funct. Theory 2 (2002), no. 1, 257–265.
  • J. L. Schiff, Normal families, Universitext, Springer, New York, 1993.
  • W. Schwick, Exceptional functions and normality, Bull. London Math. Soc. 29 (1997), no. 4, 425–432.
  • L. Yang, Normality for families of meromorphic functions, Sci. Sinica Ser. A 29 (1986), no. 12, 1263–1274.
  • L. Yang, Value distribution theory, translated and revised from the 1982 Chinese original, Springer, Berlin, 1993.