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NORMALITY AND SHARED SETS

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  01 June 2009

MINGLIANG FANG  and
LAWRENCE ZALCMAN
MINGLIANG FANG
Affiliation:
Institute of Applied Mathematics, South China Agricultural University, Guangzhou, 510642, PR China (email: hnmlfang@hotmail.com)
LAWRENCE ZALCMAN*
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel (email: zalcman@macs.biu.ac.il)
*
✉For correspondence; e-mail: zalcman@macs.biu.ac.il
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Abstract

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Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

Keywords

meromorphic functionnormalityshared set

MSC classification

Secondary: 30D45: Bloch functions, normal functions, normal families
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 3 , June 2009 , pp. 339 - 354
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The research of the first author was supported by the NNSF of China (Grant No. 10771076) and the NSF of Guangdong Province, China. The research of the second author was supported by the German–Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003.

References

[1]Bergweiler, W., ‘Normality and exceptional values of derivatives’, Proc. Amer. Math. Soc. 129 (2001), 121–129.CrossRefGoogle Scholar
[2]Bureau, F., ‘Mémoire sur les fonctions uniformes à point singulier essentiel isolé’, Mém. Soc. Roy. Sci. Liège (3) 17 (1932).Google Scholar
[3]Chen, H. H. and Lappan, P., ‘Spherical derivative, higher derivatives and normal families’, Adv. Math. (China) (6) 25 (1996), 517–524.Google Scholar
[4]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
[5]Fang, M. L. and Zalcman, L., ‘Normal families and shared values of meromorphic functions’, Ann. Polon. Math. 80 (2003), 133–141.CrossRefGoogle Scholar
[6]Fang, M. L. and Zalcman, L., ‘Normal families and shared values of meromorphic functions III’, Comput. Methods Funct. Theory 2 (2002), 385–395.CrossRefGoogle Scholar
[7]Fang, M. L. and Zalcman, L., ‘A note on normality and shared values’, J. Aust. Math. Soc. 76 (2004), 141–150.CrossRefGoogle Scholar
[8]Frank, G., ‘Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen’, Math. Z. 149 (1976), 29–36.CrossRefGoogle Scholar
[9]Gu, Y. X., ‘Un critère de normalité des familles de fonctions méromorphes’, Sci. Sinica, Special Issue I (1979), 267–274.Google Scholar
[10]Hayman, W. K., Meromorphic Functions (Clarendon Press, Oxford, 1964).Google Scholar
[11]Hiong, K. L., ‘Sur les fonctions holomorphes dont les dérivées admettant une valeur exceptionelle’, Ann. École Norm. Sup. (3) 72 (1955), 165–197.CrossRefGoogle Scholar
[12]Langley, J. K., ‘Proof of a conjecture of Hayman concerning f and f′′’, J. London Math. Soc. (2) 48 (1993), 500–514.CrossRefGoogle Scholar
[13]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
[14]Pang, X. C. and Zalcman, L., ‘Normal families and shared values’, Bull. London Math. Soc. 32 (2000), 325–331.CrossRefGoogle Scholar
[15]Schiff, J., Normal Families (Springer, Berlin, 1993).CrossRefGoogle Scholar
[16]Sun, D. C., ‘The shared value criterion for normality’, J. Wuhan Univ. Natur. Sci. 3 (1994), 9–12.Google Scholar
[17]Wang, Y. F. and Fang, M. L., ‘Picard values and normal families of meromorphic functions with multiple zeros’, Acta Math. Sinica, New Series 14 (1998), 17–26.Google Scholar
[18]Yang, L., Value Distribution Theory (Springer, Berlin, 1993).Google Scholar
[19]Zalcman, L., ‘Normal families: new perspectives’, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215–230.CrossRefGoogle Scholar