## Nagoya Mathematical Journal

### Normality, quasinormality and periodic points

Jianming Chang

#### Abstract

Let $M \ge 1$ be a positive number. Let $\mathcal{F}$ be a family of holomorphic functions $f$ in some domain $D \subset \mathbb{C}$ for which there exists an integer $k = k(f) \ge 2$ such that $|(f^{k})'(\zeta)| \le M^{k}$ for every periodic point $\zeta$ of period $k$ of $f$ in $D$. We show first that $\mathcal{F}$ is quasinormal of order at most one in $D$. This strengthens a result of W. Bergweiler. Secondly, for the case $M = 1$, we prove that $\mathcal{F}$ is normal in $D$ if there exists a positive number $K < 3$ such that $|f'(\eta)| \le K$ for each $f \in \mathcal{F}$ and every fixed point $\eta$ of $f$ in $D$. This improves a result of M. Essén and S. J. Wu. We also construct an example which shows that the condition $|f'(\eta)| \le K < 3$ can not be replaced by $|f'(\eta)| < 3$.

#### Article information

Source
Nagoya Math. J., Volume 195 (2009), 77-95.

Dates
First available in Project Euclid: 14 September 2009

https://projecteuclid.org/euclid.nmj/1252934372

Mathematical Reviews number (MathSciNet)
MR2552954

Zentralblatt MATH identifier
1185.30033

#### Citation

Chang, Jianming. Normality, quasinormality and periodic points. Nagoya Math. J. 195 (2009), 77--95. https://projecteuclid.org/euclid.nmj/1252934372

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