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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 \({\mathcal F}\) 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 \(k=1\).

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