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Normal families and shared values. (English) Zbl 1030.30031
Let $$D$$ be a domain in $$\mathbb C$$. For a meromorphic function $$f$$ in $$D$$ and $$a \in \mathbb C$$ 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 W. Schwick [Arch. Math. 59, 50-54 (1992; Zbl 0758.30028)] states that if $$\mathcal{F}$$ is a family of meromorphic functions in $$D$$ such that $$f$$ and $$f'$$ share three distinct values $$a_1$$, $$a_2$$, $$a_3 \in \mathbb C$$ for every $$f \in \mathcal{F}$$, then $$\mathcal{F}$$ 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 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 $$\mathcal{F}$$ be a family of meromorphic functions in $$D$$, all of whose zeros are of multiplicity at least $$k$$. If there exist $$b \in \mathbb C \setminus \{0\}$$ and $$h>0$$ such that for every $$f \in \mathcal{F}$$, $$\overline{E}_f(0)=\overline{E}_{f^{(k)}}(b)$$ and $$0<|f^{(k+1)}(z)|\leq h$$ for all $$z \in \overline{E}_f(0)$$, then $$\mathcal{F}$$ 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 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 Normal functions of one complex variable, normal families 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
##### Keywords:
meromorphic functions; shared values; normal families
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