高阶线性微分方程解的零点分布及Zygmund-型空间

Zero Distribution of Solutions of Higher-Order Linear Differential Equations and Zygmund Type Space

中文关键词:  线性微分方程  一致分离序列  Zygmund-型空间

英文关键词:linear differential equation  uniformly separated sequence  Zygmund type space

中文摘要:

      本文主要考虑以下两个问题: (1) 建立非齐次线性微分方程$$f'''+A_2(z)f''+A_1(z)f'+A_0(z)f=A_3(z),$$ 系数增长性与解的零点的几何分布的相互关系, 其中 $A_0(z),\ldots, A_3(z)$为单位圆内的解析函数; (2) 找到一些使方程$$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+A_0(z)f=0,$$ 所有解属于Zygmund-型空间的充分条件. 我们得到的结果推广了Heittokangas, Gr\"{o}hn, Korhoneon 和 R\"{a}tty\"{a}的部分结果.

英文摘要:

      The aim of this paper is to consider the following two problems: (1)~~Establish interrelationships between the growth of coefficients and the geometric distribution of zeros of solutions of non-homogeneous linear differential equation $$f'''+A_2(z)f''+A_1(z)f'+A_0(z)f=A_3(z),$$ where $A_0(z),\ldots, A_3(z)$ are analytic functions in the unit disc $\mathbb{D}$; (2)~~Find some sufficient conditions on the analytic coefficients of the differential equation $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+A_0(z)f=0,$$ for all solutions to belong to the Zygmund type space. The results we obtain are a generalization of some earlier results by Heittokangas, Gr\"{o}hn, Korhoneon and R\"{a}tty\"{a}.