Kodai Mathematical Journal

Growth of solutions of an $n$-th order linear differential equation with entire coefficients

Abstract

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 $\left( \left[ 3\right] ,\text{ }% p.\text{ }418\right) .$

Article information

Source
Kodai Math. J., Volume 25, Number 3 (2002), 240-245.

Dates
First available in Project Euclid: 17 December 2003

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1071674457

Digital Object Identifier
doi:10.2996/kmj/1071674457

Mathematical Reviews number (MathSciNet)
MR2003h:34182

Zentralblatt MATH identifier
1072.34515

Citation

Bela\"{i}di, Benharrat; Hamouda, Saada. Growth of solutions of an $n$-th order linear differential equation with entire coefficients. Kodai Math. J. 25 (2002), no. 3, 240--245. doi:10.2996/kmj/1071674457. https://projecteuclid.org/euclid.kmj/1071674457