关于亚纯代数体函数的奇异方向

数学物理学报 ›› 2009, Vol. 29 ›› Issue (6): 1642-1650.

关于亚纯代数体函数的奇异方向

南阳师范学院数学与统计学院河南南阳 473061

收稿日期:2008-09-04 修回日期:2009-10-15 出版日期:2009-12-25 发布日期:2009-12-25

On Singular Directions of |Meromorphic Algebroidal Functions

School of Mathematics and Statistics, Nanyang Normal University, Henan Nanyang 473061

Received:2008-09-04 Revised:2009-10-15 Online:2009-12-25 Published:2009-12-25

摘要/Abstract

摘要：

设T(r, w)满足: lim_{r →∞}lg T(r, w)/lg r =0, lim_r →∞lg T(r, w)/lg lg r =+ ∞, 则一定存在一条方向arg z=θ₀ ,使对任意给定N>0,任意复数 a (至多有2 v个例外值), 有∑_i 1/(lg|z_i(a; ?(θ₀, δ))|)^N=∞. 设T(r, w)满足: lim_r →∞T(r, w)/lg^Kr =+∞, lim_r →+∞lg T(r, w) /lg lg r =M, 则一定存在一条方向argz= θ₀ ,对任意复数a (至多有2 v个例外值), 有∑_i1/lg|z_i(a; ?(θ₀, δ))|)^σ=∞(σ = M-2或σ = M-2-ε. 即使在亚纯函数,这些奇异方向也未见研究.

关键词: 亚纯代数体函数, 奇异方向

Abstract:

Let w(z) be a meromorphic algebroidal function for |z|<+∞. If lim_r →∞lg T(r, w)/lg r =0, lim_r →∞lg T(r, w)/lg lg r =+∞, then there exists a direction arg z=θ₀ such that, for any N>0, ∑_i1/(lg|z_i(a; ?(θ₀, δ))|)^N=+∞, with 2 v possible exceptions for a. If lim_r _→∞T(r, w)/lg^Kr =+∞, lim_r _→∞lg T(r, w)/lg lg r =M, then there exists a direction arg z=θ₀ such that ∑_i1/lg|z_i(a; ?(θ₀, δ))|)^σ=∞ (σ=M-2 or σ=M-2-ε), with 2 v possible exceptions for a. For meromorphic functions, these singular directions have not been studied yet.

Key words: Meromorphic algebroidal function, Singular direction

中图分类号:

30D30

张洪申. 关于亚纯代数体函数的奇异方向[J]. 数学物理学报, 2009, 29(6): 1642-1650.

ZHANG Hong-Shen. On Singular Directions of |Meromorphic Algebroidal Functions[J]. Acta mathematica scientia,Series A, 2009, 29(6): 1642-1650.

参考文献

[1] Lu Yinian. On the Borel directions of meromorphic algebroidal function. Science in China(in Chinese), 1981, A(6): 675--680

[2] Lu Yinian, Gu Yongxing. On the existence of Borel directions of meromorphic algebroidal function. Chinese Science Bulletin(in Chinese), 1983, 3(5): 264--266

[3] Lu Qian, Gu Yongxing. On Nevanlinna directions of algebroid functions. Acta Mathematica Scientia, 2005, 25B(2): 367--375

[4] Sun Daochun. The existence theorem of the Nevanlinna directions. Chinese Annals of Mathematics(in Chinese), 1986, 7A(2): 212--221

[5] Gao Zongshen, Sun Daochun. On the maximum type Borel direction of meromorphic algebroidal function. Chinese Annals of Mathematics(in Chinese), 1997, 18A(6): 701--710

[6] Zhang Hongshen, Sun Daochun. Fill-up discs of quasi-meromorphic mappings. Acta Mathematica Scientia(in Chinese), 2004, 24A(6): 758--766

[7] Toda N. Sur les directions de Julia et de Borel des functions algebroides. Nagoya Math Journal, 1969, 34: 1--23

[8] Lu Yinian. Julia directions of meromorphic function and meromorphic algebroidal function. Acta Mathematica Sinica(in Chinese), 1984, 27(6): 367--373

[9] Tsuji M. Potential Theory in Modern Function Theory. Tokyo: Maruzen, 1959: 271--275

[10] Yang Le. Value Distribution and New Studies. Beijing: Academic Press(in Chinese), 1982: 96--97

[11] He Yuzan, Xiao Xiuzhi. Algebroidal Function and Ordinary Differential Equations. Beijing: Academic Press(in Chinese), 1988: 5--26

[12] GanHuilin, Sun Daochun. The maximality Borel direction of zero order algebroidal function. Acta Mathematica Scientia(in Chinese), 2005, 25A(5): 673--677

[13] 张洪申. 单位圆内拟亚纯映射的重值.数学物理学报, 2006, 26A(3): 337--342

[14] 吴昭君, 孙道椿. 代数体函数的一类新的奇异方向.数学物理学报, 2008, 28A(5): 879--885

[15] 柳学坤, 吴昭君. 关于亚纯函数的奇异方向. 数学物理学报, 2007, 27A(5): 590--597