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Normal families and shared values. (English) Zbl 1030.30031
Let $D$ be a domain in $\Bbb C$. For a meromorphic function $f$ in $D$ and $a \in \Bbb C$ let $$ \overline{E}_f(a) = \{ z \in D : f(z)=a \} . $$ Two meromorphic functions $f$ and $g$ in $D$ are said to share the value $a$ if $\overline{E}_f(a)=\overline{E}_g(a)$. A result of {\it W. Schwick} [Arch. Math. 59, 50-54 (1992; Zbl 0758.30028)] states that if $\Cal{F}$ is a family of meromorphic functions in $D$ such that $f$ and $f'$ share three distinct values $a_1$, $a_2$, $a_3 \in \Bbb C$ for every $f \in \Cal{F}$, then $\Cal{F}$ is normal in $D$. The corresponding statement in which $f'$ is replaced by $f^{(k)}$ $(k \geq 2)$ is no longer true. A counterexample was given by {\it G. Frank} and {\it W. Schwick} [N. Z. J. Math. 23, 121-123 (1994; Zbl 0830.30019)]. In this paper the authors prove the following result. Theorem. Let $\Cal{F}$ be a family of meromorphic functions in $D$, all of whose zeros are of multiplicity at least $k$. If there exist $b \in \Bbb C \setminus \{0\}$ and $h>0$ such that for every $f \in \Cal{F}$, $\overline{E}_f(0)=\overline{E}_{f^{(k)}}(b)$ and $0<|f^{(k+1)}(z)|\leq h$ for all $z \in \overline{E}_f(0)$, then $\Cal{F}$ 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 $\overline{E}_f(0) \subset \overline{E}_{f'}(b)$ and that $|f''(z)|\leq h$ for $z \in \overline{E}_{f'}(b)$. In the special case $\overline{E}_f(0)=\emptyset$, the above theorem gives a result of {\it 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 {\it 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)].

30D45Bloch functions, normal functions, normal families
30D35Distribution of values (one complex variable); Nevanlinna theory
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