# zbMATH — the first resource for mathematics

Improvement of Marty’s criterion and its application. (English) Zbl 0777.30018
The classical spherical type criterion of F. Marty states that a family $${\mathcal F}$$ of functions meromorphic in a domain $$D$$ is a normal family iff a spherical estimate $$| f'(z)|\leq c_ K(1+| f(z)|^ 2)$$ holds uniformly for $$f\in{\mathcal F}$$ 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 $${\mathcal F}$$ of meromorphic functions is normal if each function $$f\in{\mathcal F}$$ has only poles of degree at least $$k+2$$ and satisfies $$f^{(k)}-af^ 3\neq b$$ everywhere, where $$a$$ and $$b$$ are fixed complex numbers. This result was established by D. Drasin for holomorphic functions where $$k=1$$.

##### MSC:
 30D45 Normal functions of one complex variable, normal families 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
normal family