Normality, quasinormality and periodic points

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 $|f'(\eta)| < 3$.