主要证明了: 设k≥2是一个正整数, M是一个正数,c是一个非零有穷复数, F是区域D内的一族亚纯函数,其中每个函数的零点的重数至少是k. 若对于F中的任意函数f, f(z)=0⇔f(k)(z)=0,f(k)(z)=c⇒|f(k+1)(z)|≤M, 则F在D内正规, 其中c≠0是必需的.
Abstract
It is mainly proved: let k ≥ 2 be a positive integer, M a positive number, let c (≠0) be a finite value, and let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity k at least. If, for each function f ∈ F, f(z)=0⇔f(k)(z)=0, f(k)(z)=c⇒|f(k+1)(z)| ≤ M, then F is normal in D. And c ≠ 0 is necessary.
关键词
亚纯函数 /
分担值 /
正规族
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Key words
meromorphic functions /
shared values /
normal families
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参考文献
[1] Hayman W. K., Meromorphic Functions, Clarendon Press, Oxford, 1964.
[2] Yang L., Value Distribution Theory, Springer-Verlag, Berlin, 1993.
[3] Gu Y. X., Pang X. C., Fang M. L., The Theory of Normal Families and Its Application (in Chinese), Science Press, Beijing, 2007.
[4] Schwick W., Sharing values and normality, Arch. Math., 1992, 59: 50-54.
[5] Fang M. L., Zalcman L., Normal families and shared values of meromorphic functions III, Comput. Methods Funct. Theory, 2002, 2(2): 385-395.
[6] Chang J. M., Normality concerning shared values (in Chinese), Sci. Sin. Math., 2009, 39(4): 399-404.
[7] Huang X. J., Liu Z. X., A normal criterion concerning shared values (in Chinese), Acta Math. Sci. Ser. A Chin. Ed., 2011, 31(2): 540-545.
[8] Fang M. L., Zalcman L., A note on normality and shared values, J. Aust. Math. Soc., 2004, 76: 141-150.
[9] Zalcman L., Normal families: New perspectives, Bull. Amer. Math. Soc., 1998, 35: 215-230.
[10] Pang X. C., Zalcman L., Normal families and shared values, Bull. London Math. Soc., 2000, 32: 325-331.
[11] Bergweiler W., Langley J. K., Nonvanishing derivatives and normal families, J. Anal. Math., 2003, 91: 353-367.
[12] Fang M. L., Picard values and normality criterion, Bull. Korean Math. Soc., 2001, 38(2): 379-387.
[13] Frank G., Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z., 1976, 149: 29-36.
[14] Langley J. K., Proof of a conjecture of Hayman concerning f and f, J. London Math. Soc., 1993, 48(2): 500-514.
[15] Meng D. W., Hu P. C., Normality criteria of meromorphic functions sharing one value, J. Math. Anal. Appl., 2011, 381: 724-731.
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脚注
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