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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, $$\sup\{(1-|z|^2)f^{\#}(z):z \in D, f \in \mathcal F\} < \infty$$.
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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