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On the normal meromorphic functions. (English) Zbl 1132.30339
Summary: Let \(\mathcal F\) be a family of functions meromorphic in \(D\) such that all the zeros of \(f\in\mathcal F\) are of multiplicity at least \(k\) (a positive integer), and let \(E\) be a set containing \(k+4\) points of the extended complex plane. If, for each function \(f\in\mathcal F\), there exists a constant \(M\) and such that \((1-|z|^2)^k |f^{(k)}(z)|/(1+|f(z)|^{k+1})\leq M\) whenever \(z\in \{f(z) \in E, z \in D\}\), then \(\mathcal F\) is a uniformly normal family in \(D\), that is, \(\sup\{(1-|z|^2)f^{\#}(z):z \in D, f \in \mathcal F\} < \infty\).
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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