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Improvement of Marty’s criterion and its application. (English) Zbl 0777.30018
The classical spherical type criterion of F. Marty states that a family $ℱ$ of functions meromorphic in a domain $D$ is a normal family iff a spherical estimate $|{f}^{\text{'}}\left(z\right)|\le {c}_{K}{\left(1+|f\left(z\right)|}^{2}\right)$ holds uniformly for $f\in ℱ$ and points $z$ in any compact subset $K\subset D$. A Euclidean type criterion of an entirely different nature is originally due to A. J. Lohwater and C. Pommerenke [Ann. Acad. Sci. Fenn., Ser. A I 550, 12 p. (1973; Zbl 0275.30027)] with a later treatment given by L. Zalcman [Am. Math. Mon. 82, 813-817 (1975; Zbl 0315.30036)]. The present paper considers similar criterion for normality of families of functions whose zeros are of degree at least $k$, where $k$ is a positive integer. As an example of how these results can be applied, the authors prove that a family $ℱ$ of meromorphic functions is normal if each function $f\in ℱ$ has only poles of degree at least $k+2$ and satisfies ${f}^{\left(k\right)}-a{f}^{3}\ne b$ everywhere, where $a$ and $b$ are fixed complex numbers. This result was established by D. Drasin for holomorphic functions where $k=1$.

##### MSC:
 30D45 Bloch functions, normal functions, normal families 30D35 Distribution of values (one complex variable); Nevanlinna theory
normal family