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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 of functions meromorphic in a domain D is a normal family iff a spherical estimate |f ' (z)|c K (1+|f(z)| 2 ) holds uniformly for f and points z in any compact subset KD. 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 of meromorphic functions is normal if each function f has only poles of degree at least k+2 and satisfies f (k) -af 3 b everywhere, where a and b are fixed complex numbers. This result was established by D. Drasin for holomorphic functions where k=1.

MSC:
30D45Bloch functions, normal functions, normal families
30D35Distribution of values (one complex variable); Nevanlinna theory
Keywords:
normal family