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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
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