一类整函数系数线性微分方程解的增长性

Growth of Solutions of Certain Linear Differential Equations with Entire Coefficients

本文主要研究某类整函数系数高阶线性微分方程解的增长性,这类方程有一个系数为满足Denjoy猜想极值情况的整函数.运用亚纯函数值分布理论和整函数的渐近值理论,通过比较方程中每一项的模的大小,得到这类方程解的增长级的估计.对只有一个系数起控制作用的方程,当其存在一个系数为二阶微分方程的解时,得到上述方程的非零解都为无穷级.对系数具有相同增长级的方程,当其系数具指数函数形式时,得到上述方程的非零解也为无穷级.文中所得结果是对线性微分方程相关结果的推广和补充.

This paper is devoting to study the growth of solutions of some types of higherorder linear differential equations with entire coefficients.One of its coefficients is an entire function extremal for Denjoy’s conjecture.By using the value distribution theory of meromorphic functions and the asymptotic value theory of entire functions,and comparing the size of the module of each item appeared in such equations,the estimation on the growth order of its solutions are obtained.It is proved that any nontrivial solution of the equation with one dominant coefficient is of infinite,when there exists one coefficient satisfying the second-order differential equation.The same result also holds for the equation with coefficients having the same growth order and the exponential expressions.The obtained results are the generalization and supplement of some previous results in linear differential equations.