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A note on normality and shared values

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

Mingliang Fang  and
Lawrence Zalcman
Mingliang Fang
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing 210097 P. R., China, e-mail: mlfang@pine.njun.edu.cn
Lawrence Zalcman
Affiliation:
Department of Mathematics and Statistics, Bar-Ilan University52900 Ramat-Gan, Israel e-mail: zalcman@macs.biu.ac.il
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Abstract

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Let k be a positive integer and b a nonzero constant. Suppose that F is a family of meromorphic functions in a domain D. If each function f ∈ F has only zeros of multiplicity at least k + 2 and for any two functions f, g ∈ F, f and g share 0 in D and f(k) and g(k) share b in D, then F is normal in D. The case f ≠ 0, f(k) ≠ b is a celebrated result of Gu.

Keywords

meromorphic functionnormalityshared value

MSC classification

Secondary: 30D45: Bloch functions, normal functions, normal families
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 1 , February 2004 , pp. 141 - 150
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Fang, M. L. and Hong, W., ‘Some results on normal family of meromorphic functions’, Bull. Malays. Math. Sci. Soc. (2) 23 (2000), 143–151.Google Scholar
[2]Gu, Y. X., ‘Un critère de normalité de fonctions méromorphes’, Sci. Sinica, Special Issue 1 (1979), 267–274.Google Scholar
[3]Hayman, W. K., ‘Picard values of meromorphic functions and their derivatives’, Ann. of Math. (2) 70 (1959), 9–42.CrossRefGoogle Scholar
[4]Hayman, W. K., Meromorphic functions (Clarendon Press, Oxford, 1964).Google Scholar
[5]Milloux, H., Les fonctions méromorphes et leurs dérivées (Hermann et Cie., Paris, 1940).Google Scholar
[6]Montel, P., ‘Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine’, Ann. École Norm. Sup. (3) 29 (1912), 487–535.CrossRefGoogle Scholar
[7]Schiff, J., Normal families (Springer, 1993).CrossRefGoogle Scholar
[8]Sun, D. C., ‘The shared value criterion for normality’, J. Wuhan Univ. Natur. Sci. Ed. 3 (1994), 9–12.Google Scholar
[9]Wang, Y. F. and Fang, M. L., ‘Picard values and normal families of meromorphic functions with multiple zeros’, Acta Math. Sinica (N.S.) 14 (1998), 17–26.Google Scholar
[10]Xue, G. F. and Pang, X. C., ‘A criterion for normality of a family of meromorphic functions’, J. East China Norm. Univ. Natur. Sci. Ed. 2 (1988), 15–22.Google Scholar
[11]Yang, L., ‘Normality for families of meromorphic functions’, Sci. Sinica. Ser. A 29 (1986), 1263–1274.Google Scholar
[12]Yang, L., Value distribution theory (Springer, Berlin; Science Press, Beijing, 1993).Google Scholar
[13]Zalcman, L., ‘Normal families: New perspectives’, Bull. Amer. Math. Soc. 35 (1998), 215–230.CrossRefGoogle Scholar