We consider a differential equation $f^{\left( n\right)
}+A_{n-1}\left( z\right) f^{\left( n-1\right) }+...+A_{1}\left( z\right)
f^{^{/}}+A_{0}\left( z\right) f=0,$ where $A_{0}\left( z\right)
,...,A_{n-1}\left( z\right) $ are entire functions with $A_{0}\left(
z\right) \hbox{$/\hskip
-11pt\equiv$}0$. Suppose that there exist a positive number $\mu ,$\ and a
sequence $\left( z_{j}\right) _{j\in N}$ with $\stackunder{j\rightarrow
+\infty }{\lim }z_{j}=\infty ,$ \ and also two real numbers $\alpha ,\beta $
$\left( \ 0\leq \beta \alpha \right) $\ such that \ $\left| A_{0}\left(
z_{j}\right) \right| \geq e^{\alpha \left| z_{j}\right| ^{\mu }}\quad $and$%
\quad \left| A_{k}\left( z_{j}\right) \right| \leq e^{\beta \left|
z_{j}\right| ^{\mu }}$ as $\ j\rightarrow +\infty $ $\left(
k=1,...,n-1\right) $. We prove that all solutions \ $f%
\hbox{$/\hskip
-11pt\equiv$}0$ of this equation are of infinite order. This result is a
generalization of one theorem of Gundersen $\left( \left[ 3\right] ,\text{ }%
p.\text{ }418\right) .$