On the growth of solutions of some higher order linear differential equations with entire coefficients

Abstract

In this paper, we investigate the order and the hyper-order of entire solutions of the linear differential equation \begin{equation*} f^{\left( k\right) }+\left( D_{k-1}+B_{k-1}e^{b_{k-1}z}\right) f^{\left(k-1\right) }+ ... +\left( D_{1}+B_{1}e^{b_{1}z}\right) f^{\prime }+\left( D_{0}+A_{1}e^{a_{1}z}+A_{2}e^{a_{2}z}\right) f=0 \end{equation*} where $A_{j}\left( z\right) $ $\left( \not\equiv 0\right) $ $(j=1,2)$, $ B_{l}\left( z\right) $ $\left( \not\equiv 0\right) $ $(l=1,...,k-1)$, $D_{m}$ $(m=0,...,k-1)$ are entire functions with $\max \{\sigma \left( A_{j}\right), \sigma \left( B_{l}\right), \sigma \left( D_{m}\right) \}<1$, $a_{1}$, $ a_{2}$, $b_{l}$ $(l=1,...,k-1)$ are complex numbers. Under some conditions, we prove that every solution $f\left( z\right) \not\equiv 0$ of the above equation is of infinite order and with hyper-order 1.

Keywords

• linear differential equations
• entire solutions
• order of growth
• hyper-order