一类高阶微分方程的亚纯解与其小函数的复振荡

摘要

In this paper, by applying the Nevanlinna value distribution theory and methods of meromorphic functions, we investigate the complex oscillation of a class of higher order homo-geneous linear differential equations with meromorphic coeffcients, study the relation between the meromorphic solutions of the higher order homogeneous linear differential equation with meromorphic coeffcients and functions of smaller growth, investigate the relation between the first derivatives of the meromorphic solutions of the higher order homogeneous linear differen-tial equation with meromorphic coeffcients and functions of smaller growth, and obtain some estimations between the meromorphic solution and the first order derivatives of the meromor-phic solution of the higher order homogeneous linear differential equation with meromorphic coeffcients and functions of smaller growth. These results has generalized and improved some results in previous references. Finally, the examples are given to show that our results are precise.%利用亚纯函数的Nevanlinna基本理论和方法，本文研究了一类高阶齐次线性微分方程亚纯解的复振荡、亚纯系数的高阶齐次线性微分方程亚纯解与小函数的关系、以及亚纯系数的高阶齐次线性微分方程亚纯解的一阶导数与小函数的关系，得到了亚纯系数高阶齐次线性微分方程的亚纯解和其导数取小函数的精确估计，推广了一些已有文献的结论，得到了更一般，更精确的结果，且文中有例子表明结果是精确的。