正规族与分担函数

刘晓毅, 程春暖

数学学报 ›› 2013, Vol. 56 ›› Issue (6) : 941-950.

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数学学报 ›› 2013, Vol. 56 ›› Issue (6) : 941-950. DOI: 10.12386/A2013sxxb0090
论文

正规族与分担函数

    刘晓毅1, 程春暖2
作者信息 +

Normal Families and Shared Functions

    Xiao Yi LIU1, Chun Nuan CHENG2
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文章历史 +

摘要

设a(≠ 0)和b( 0)是区域D上的两个全纯函数,并且在函数b的零点处a'不取0.F={f}是区域D内的一族亚纯函数,若每个f的零点都是重级的,且满足f = a⇔f' =b,则F在区域D内正规.举例说明结论成立的条件是必不可少的.

Abstract

Let a and b be holomorphic functions on a plane domain D such that a has no zero, b is not zero identically, and a does not take the value 0 at the zeros of b. Let F = {f} be a family of meromorphic functions on D, all of whose zeros are multiple, such that f = a ⇔ f = b. Then the family F is normal on D. Examples are given to show the necessity of the conditions.

关键词

亚纯函数 / 正规族 / 分担值 / 分担函数

Key words

meromorphic functions / normal families / shared values / shared functions

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刘晓毅, 程春暖. 正规族与分担函数. 数学学报, 2013, 56(6): 941-950 https://doi.org/10.12386/A2013sxxb0090
Xiao Yi LIU, Chun Nuan CHENG. Normal Families and Shared Functions. Acta Mathematica Sinica, Chinese Series, 2013, 56(6): 941-950 https://doi.org/10.12386/A2013sxxb0090

参考文献

[1] Bergweiler W., Pang X. C., On the derivative of meromorphic functions with multiple zeros, J. Math. Anal. Appl., 2003, 278: 285-292.

[2] Chen H. H., Gu Y. X., Improvement of Marty's criterion and its applications, Sci. China, Ser. A, 1993, 36: 674-681.

[3] Fang M. L., Zalcman L., Normal families and shared values Ⅲ, Comput. Methods Funct., 2002, 1: 385-395.

[4] Hayman W. K., Meromorphic Functions, Clarendon Press, Oxford, 1964.

[5] Lei C. L., Yang D. G., Fang M. L., Normal families and shared values of Meromorphic functions, J. Math. Anal. Appl., 2010, 364: 143-150.

[6] Pang X. C., Zalcman L., Normal families and shared values, Bull. London Math. Soc., 2000, 32: 325-331.

[7] Schiff J. L., Normal Families, Springer-Verlag, New York, 1993.

[8] Schwick W., Sharing values and normaltity, Arch. Math., 1992, 59: 50-54.

[9] Wang Y. F., Fang M. L., Picard values and normal families of meromorphic functions with multiple zeros, Acta Mathematica Sinica, New Series, 1998, 14(1): 17-26.

[10] Yang L., Value Distribution Theory, Spring-Verlag, Berlin, 1993.

基金

国家自然科学基金资助项目(11171045)

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