Computational Methods and Function Theory 1 (2001), No. 2, 375--386
Copyright Heldermann Verlag 2001
Shahar Nevo
shahar@wisdom.weizmann.ac.il, Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel.
A family $\mathcal{F}$ of meromorphic functions on a plane domain $D$ is called quasi-normal on $D$ if each sequence $S$ of functions in $\mathcal{F}$ has a subsequence which converges locally $\chi$-uniformly on $D\setminus E$, where $E=E(S)$ is a subset of $D$ having no accumulation points in $D$. The notion of quasi-normality was introduced by Montel. It generalizes the concept of normal family, which corresponds to $E=\emptyset$. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family $\mathcal{F}$ of meromorphic functions on a plane domain $D$ is $Q_m$-normal $(m=0,1,2,\dots)$ if each sequence $S$ of functions in $\mathcal{F}$ has a subsequence which converges locally $\chi$-uniformly on the domain $D\setminus E$, where $E=E(S)\subset D$ satisfies $E_D^{(m)}=\emptyset$. $m$ is said to be the degree of $\mathcal{F}$. (Here $E_D^{(m)}$ is the $m$-th derived set of $E$ in $D$.) In particular, a $Q_0$-normal family is a normal family, and a $Q_1$-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let $f$ be a nonconstant meromorphic function on $\mathbb{C}$, and write $f_n(z)=f(nz)$. Then the family $\Pi(f)=\{f_n:n\in\mathbb{N}\}$ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if $f$ is a rational function, then $\Pi(f)$ is quasi-normal of exact order $1$. In the opposite direction, if $f$ is not rational, and there exist $a,b\in\hat{\mathbb{C}}$ such that the equations $f(z)=a$ and $f(z)=b$ have only a finite number of solutions in~$\mathbb{C}$, then $\Pi(f)$ fails to be $Q_m$-normal for any value of $m$.
Keywords: Normal family.
MSC 2000: 30D45, 30D20, 30D30.
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