关于一个亚纯函数与其各级导数的公共Borel方向

北京大学学报（自然科学版）

关于一个亚纯函数与其各级导数的公共Borel方向

庄圻泰

北京大学数学系，北京，100871

收稿日期:1995-09-08 出版日期:1997-01-20 发布日期:1997-01-20

On Common Borel Directions of a Meromorphic Function and its Derivatives

CHUANG Chitai

Department of Mathematics, Peking University, Beijing, 100871

Received:1995-09-08 Online:1997-01-20 Published:1997-01-20

摘要/Abstract

摘要： 证明了下列定理: 设 f(z)为一有穷正级λ(0<λ<+∞)的亚纯函数, 并设L: argz=θ₀为一方向。假定任给二数δ(0<δ<1)及ε(0<ε<λ), 恒可得一数r₀使对于每一数r>r₀,集合{z || z-re^iθ₀|<δr, |f(z)|≤e^{r^λ-ε}}不能范围在有穷个圆|z-z_j|<ρ_i(j=1,2,...,p),Σ^p_j=1ρ_j≤e^{-r^ε}中,则下列二结论成立:
1) 若对于一整数m≥1, L为f^(m)(z)的一个λ级Borel方向, 则L为f(z)的一个λ级Borel方向。
2) 若L为f(z)的一个λ级Borel 方向, 则L为f(z)和所有各级导数 f^(m)(z) (m=1,2,...)的一个公共λ级Borel方向。

关键词: Borel方向, 亚纯函数, 各级导数

Abstract: Since Valiron proposed the problem of studying the common Borel directions of an entire or meromorphic function and its derivative, it had been studied by various authors such as Cartwright, Rauch, Chuang, Milloux, Chang, Yang and Hu. Recently Zhu also studies this problem. His main result is that a Borel direction of order lambda of a meromorphic function f(z) of finite positive order lambda is a Borel direction of order lambda of one at least of the three functions f'(z), (zf(z))' and ((z+1) f(z))'. In the proof he skillfully applies a theorem of Chuang on the growth of meromorphic functions and a theorem of Yang on the value-distribution of such functions. In the present paper we study the problem in a different way. Also basing upon these two theorems, we shall show that, under certain conditions, a given direction argz=θ₀ is a common Borel direction of order lambda of a meromorphic function f(z) of finite positive order lambda and all its successive derivatives f(m)(z) (m=1,2,…).

Key words: Borel directions, meromorphic functions, derivatives

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