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Normal families and shared values. (English) Zbl 1030.30031

Let $D$ be a domain in $ℂ$. For a meromorphic function $f$ in $D$ and $a\in ℂ$ let

${\overline{E}}_{f}\left(a\right)=\left\{z\in D:f\left(z\right)=a\right\}·$

Two meromorphic functions $f$ and $g$ in $D$ are said to share the value $a$ if ${\overline{E}}_{f}\left(a\right)={\overline{E}}_{g}\left(a\right)$. A result of W. Schwick [Arch. Math. 59, 50-54 (1992; Zbl 0758.30028)] states that if $ℱ$ is a family of meromorphic functions in $D$ such that $f$ and ${f}^{\text{'}}$ share three distinct values ${a}_{1}$, ${a}_{2}$, ${a}_{3}\in ℂ$ for every $f\in ℱ$, then $ℱ$ is normal in $D$. The corresponding statement in which ${f}^{\text{'}}$ is replaced by ${f}^{\left(k\right)}$ $\left(k\ge 2\right)$ is no longer true. A counterexample was given by G. Frank and W. Schwick [N. Z. J. Math. 23, 121-123 (1994; Zbl 0830.30019)]. In this paper the authors prove the following result.

Theorem. Let $ℱ$ be a family of meromorphic functions in $D$, all of whose zeros are of multiplicity at least $k$. If there exist $b\in ℂ\setminus \left\{0\right\}$ and $h>0$ such that for every $f\in ℱ$, ${\overline{E}}_{f}\left(0\right)={\overline{E}}_{{f}^{\left(k\right)}}\left(b\right)$ and $0<|{f}^{\left(k+1\right)}\left(z\right)|\le h$ for all $z\in {\overline{E}}_{f}\left(0\right)$, then $ℱ$ is a normal family in $D$.

The corresponding result for holomorphic functions with $k=1$ is due to X. Pang [Analysis, München 22, 175-182 (2002; Zbl 1030.30031)] and requires only ${\overline{E}}_{f}\left(0\right)\subset {\overline{E}}_{{f}^{\text{'}}}\left(b\right)$ and that $|{f}^{\text{'}\text{'}}\left(z\right)|\le h$ for $z\in {\overline{E}}_{{f}^{\text{'}}}\left(b\right)$. In the special case ${\overline{E}}_{f}\left(0\right)=\varnothing$, the above theorem gives a result of Y. Ku [Sci. Sinica 1979, Special Issue I on Math., 267-274 (1979)].

In contrast to the proofs of the above results of X. Pang and W. Schwick, the authors make no use of Nevanlinna theory. The main tool of the proof is a generalization of a version of the non-normality criterion of Z. Zalman [Am. Math. Mon. 82, 813-817 (1975; Zbl 0315.30036)] which is due to X. Pang [Sci. China, Ser. A 32, 782-791 (1989; Zbl 0687.30023)], [Sci. China, Ser. A 33, 521-527 (1990; Zbl 0706.30024)].

##### MSC:
 30D45 Bloch functions, normal functions, normal families 30D35 Distribution of values (one complex variable); Nevanlinna theory
##### Keywords:
meromorphic functions; shared values; normal families