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A Criterion for Normalcy

Published online by Cambridge University Press:  22 January 2016

Paul Gauthier
Paul Gauthier*
Affiliation:
Wayne State University
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Gavrilov [2] has shown that a holomorphic function f(z) in the unit disc |z|<1 is normal, in the sense of Lehto and Virtanen [5, p. 86], if and only if f(z) does not possess a sequence of ρ-points in the sense of Lange [4]. Gavrilov has also obtained an analagous result for meromorphic functions by introducing the property that a meromorphic function in the unit disc have a sequence of P-points. He has shown that a meromorphic function in the unit disc is normal if and only if it does not possess a sequence of P-points.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 32 , June 1968 , pp. 277 - 282
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Gauthier, P.M.: Sequences of ρ-points of meromorphic functions, Amer. Math. Soc. Notices 14 (1967), 257.Google Scholar
[2] Gavrilov, V.I.: On the distribution of values of non-normal meromorphic functions in the unit disc (Russian), Mat. Sb. 109 (n.s. 67) (1965), 408–427.Google Scholar
[3] Hille, E.: Analytic function theory, Vol. 2, Boston, 1962.Google Scholar
[4] Lange, L.H.: Sur les cercles de remplissage non-Euclidiens, Ann. Sci. École Norm. Sup. (3) 77 (1960), 257–280.Google Scholar
[5] Noshiro, K.: Cluster sets, Berlin, 1960.Google Scholar
[6] Ostrowski, A.: Über Folgen analytischer Funktionen und einige Verscharfungen des Picardschen Satzes, Math. Zeitschr. 24 (1926), 215–258.Google Scholar