涉及重级零点的有理函数值分布

Value distribution of rational function with multiple zero

摘要

设n为正整数,f(z)是复平面C内的一个亚纯函数,只有重级零点,如果对任意z ∈C,都有f′(z)≠z ⁿ成立,那么f(z)为特定形式的有理函数。

Abstract

Let n≥1 be an integer, and f(z) be a meromorphic function in complex plane, whose zeros are multiple. If f′(z)≠z ⁿ for each z ∈C, then f(z) must be restricted to specific forms of rational function.

关键词

有理函数 / 亚纯函数 / 值分布 / 重级零点

Key words

rational function / meromorphic function / value distribution / multiple zeros

参考文献

[1] NEVANLINNA R. Zur Theorie der meromorphen Funktionen[J]. Acta Mathematica, 1925, 46(1/2): 1-99.

[2] HAYMAN W K. Picard values of meromorphic functions and their derivatives[J]. Annals of Mathematics, 1959, 70(1): 9-42.

[3] WANG Yuefei, FANG Mingliang. Picard values and normal families of meromorphic functions with multiple zeros[J]. Acta Mathematica Sinica, English Series, 1998, 14(1): 17-26.

[4] PANG Xuecheng, NEVO S, ZALCMAN L. Derivatives of meromorphic functions with multiple zeros and rational functions[J]. Computational Methods and Function Theory, 2008, 8(2): 483-491.

[5] XU Yan. Normal families and exceptional functions[J]. Journal of Mathematical Analysis and Applications, 2007, 329: 1343-1354.

基金资助

国家自然科学基金资助项目(12061077)

国家自然科学基金资助项目(11961068)