Electronic Journal of Differential Equations (Apr 2014)
Growth of solutions to higher-order linear differential equations with entire coefficients
Abstract
In this article, we discuss the order and hyper-order of the linear differential equation $$ f^{(k) }+\sum_{j=1}^{k-1} (B_je^{b_jz}+D_je^{d_jz}) f^{(j) }+( A_1e^{a_1z}+A_2e^{a_2z}) f=0, $$ where $A_j(z), B_j(z), D_j(z)$ are entire functions $(\not\equiv 0)$ and $a_1,a_2,d_j$ are complex numbers $(\neq 0)$, and $b_j$ are real numbers. Under certain conditions, we prove that every solution $f\not\equiv 0$ of the above equation is of infinite order. Then, we obtain an estimate of the hyper-order. Finally, we give an estimate of the exponent of convergence for distinct zeros of the functions $f^{(j)}-\varphi $ $(j=0,1,2) $, where $\varphi$ is an entire function $(\not\equiv 0) $ and of order $\sigma (\varphi)<1$, while the solution $f$ of the differential equation is of infinite order. Our results extend the previous results due to Chen, Peng and Chen and others.
