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\markboth{\hfil Orders of solutions \hfil EJDE--2001/61}
{EJDE--2001/61\hfil Benharrat Bela\"\i di \& Saada Hamouda \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 61, pp. 1--5. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Orders of solutions of an n-th order linear differential equation
  with entire coefficients
 %
\thanks{ {\em Mathematics Subject Classifications:} 30D35, 34M10, 34C10, 34C11.
\hfil\break\indent
{\em Key words:} Linear differential equations,  entire functions, order of growth.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted July 23, 2001. Published September 17, 2001.} }
\date{}
%
\author{Benharrat Bela\"\i di \& Saada Hamouda}
\maketitle

\begin{abstract}
 We study the solutions of the differential equation
 $$f^{(n)}+A_{n-1}(z) f^{(n-1) }+\dots+A_{1}(z)f'+A_{0}(z) f=0,$$
 where the coefficients are entire functions.
 We find conditions on the coefficients so that
 every solution that is not identically zero has infinite order.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

For $n\geq 2$, we consider the linear differential equation
\begin{equation}
f^{(n) }+A_{n-1}(z) f^{(n-1)}+ \dots +A_{1}(z) f'+A_{0}(z) f=0\,,  \tag{1.1}
\end{equation}
where $A_{0}(z) ,\dots,A_{n-1}(z)$ are entire functions with
$A_{0}(z)\not\equiv 0$. Let $\rho (f)$ denote the order of the growth of
an entire function $f$ as defined in \cite{h1}:
$$
\rho(f)=\limsup_{r\to \infty}\frac{\log \big(\log\big(\max_{|z|=r}|f(z)|
\big)\big)}{\log r}\,.
$$
The value $T(r,f)=\log(\max_{|z|=r}|f(z)|)$ is known as the Nevanlinna 
characteristic of $f$ \cite{h1}.
It is well known that all solutions of (1.1) are entire functions
and when some of the coefficients of (1.1) are transcendental, (1.1) has
at least one solution with order $\rho (f) =\infty$.
The question which arises is: \begin{quote}
What conditions on $A_{0}(z) ,\dots,A_{n-1}(z)$ will guarantee
that every solution $f\not\equiv 0$ has infinite order?
\end{quote}
In this paper we prove two results concerning this question.

When $A_{0}(z) ,\dots, A_{n-1}(z)$ are polynomials with $A_{0}(z)\not\equiv 0$,
every solution of (1.1) is an entire function with finite rational order;
see for example \cite{g3}, \cite[pp. 199-209]{j1}, \cite[pp. 106-108]{v1}, and
\cite[pp. 65-67]{w1}.

In the study of the differential equation
\begin{equation}
f''+A(z) f'+B(z) f=0 \tag{1.2}
\end{equation}
where $A(z)$ and $B(z)\not\equiv 0$ are entire functions, Gundersen
proved the following results.

\begin{theorem}[{\cite[p. 418]{g1}}] \label{thmA}
Let $A(z)$  and $B(z)\not\equiv 0$ be entire functions such that for real
constants $\alpha$, $\beta$, $\theta _{1}$,  $\theta _{2}$
with $\alpha>0$, $\beta >0$, and $\theta _{1}<\theta _{2}$, we have
\begin{equation}
|B(z)| \geq \exp \{ (1+o(1)) \alpha | z| ^{\beta }\}  \tag{1.3}
\end{equation}
and
\begin{equation}
| A(z) | \leq \exp \{ o(1)| z| ^{\beta }\}  \tag{1.4}
\end{equation}
as $z\to \infty $ with $\theta _{1}\leq \arg z\leq \theta _{2}$.
Then every solution $f\not\equiv 0$  of (1.2) has infinite order.
\end{theorem}

\begin{theorem}[{\cite[p. 419]{g1}}] \label{thmB}
 Let $\{ \Phi _{k}\}$ and $\{ \theta _{k}\}$ be two finite
collections of real numbers satisfying $\Phi _{1}<\theta _{1}
<\Phi _{2}<\theta _{2}<\dots< \Phi _{n}<\theta _{n}<
\Phi _{n+1}$,  where $\Phi _{n+1}=\Phi _{1}+2\pi $,
and set
\begin{equation}
\mu =\max_{1\leq k\leq n} (\Phi _{k+1}-\theta _{k}). \tag{1.5}
\end{equation}
Suppose that $A(z) $ and $B(z) $  are entire functions such that for
some constant $\alpha \geq 0$,
\begin{equation}
| A(z) | =O(| z| ^{\alpha }) \tag{1.6}
\end{equation}
as $z\to \infty $ with $\Phi _{k}\leq \arg z\leq \theta _{k}$
 for $k=1,\dots,n$ and where $B(z) $ is transcendental with
 $\rho (B) <\frac{\pi }{\mu }$.
Then every solution $f\not\equiv 0$ of (1.2) has infinite
order.
\end{theorem}

\section{Statement and proof of results}
In this paper we prove the following two theorems:

\begin{theorem} \label{thm1}
Let $A_{0}(z),\dots, A_{n-1}(z)$, $A_{0}(z)\not\equiv 0$
be entire functions such that for real constants
$\alpha$, $\beta$, $\mu$, $\theta _{1}$, $\theta _{2}$,
where $0\leq \beta <\alpha $, $\mu >0$ and $\theta _{1}<\theta _{2}$
we have
\begin{equation}
| A_{0}(z) | \geq e^{\alpha | z| ^{\mu }} \tag{2.1}
\end{equation}
and
\begin{equation}
| A_{k}(z) | \leq e^{\beta | z|^{\mu }},\quad k=1,\dots,n-1  \tag{2.2}
\end{equation}
as $z\to \infty $ with $\theta _{1}\leq \arg z\leq \theta _{2}$.
Then every solution  $f\not\equiv 0$
of (1.1) has infinite order.
\end{theorem}

\begin{theorem} \label{thm2}
Let $\{ \Phi _{k}\}$ and $\{ \theta _{k}\} $ be two finite
collections of real numbers satisfying $\Phi _{1}<\theta _{1}<
\Phi _{2}<\theta _{2}<\dots< \Phi _{m}<\theta _{m}<
\Phi _{m+1}$  where $\Phi _{m+1}=\Phi _{1}+2\pi$,
and set
\begin{equation}
\mu =\max_{1\leq k\leq m}(\Phi _{k+1}-\theta _{k}) .\tag{2.3}
\end{equation}
Suppose that $A_{0}(z) ,\dots,A_{n-1}(z) $
are entire functions such that for some constant $\alpha \geq 0$,
\begin{equation}
| A_{j}(z) | =O(| z| ^{\alpha}),\quad j=1,\dots,n-1  \tag{2.4}
\end{equation}
as $z\to \infty $ with $\Phi _{k}\leq \arg z\leq \theta _{k}$
for $k=1,\dots,m$  and where $A_{0}(z)$ is transcendental with
$\rho (A_{0}) <\pi/\mu$.
Then every solution  $f\not\equiv 0$ of (1.1) has infinite order.
\end{theorem}

Next, we provide a lemma that is used in the proofs of our theorems.

\begin{lemma}[{\cite[p. 89]{g2}}] \label{lm1}
Let $w$ be a transcendental entire function
of finite order $\rho $. Let
$\Gamma =\{ (k_{1},j_{1}) ,(k_{2},j_{2}) ,\dots,(k_{m},j_{m})\}$
denote a finite set of distinct pairs of integers satisfying
$k_{i}>j_{i}\geq 0$ for $i=1,\dots,m$, and let
$\varepsilon >0$ be a given constant. Then there exists a set
$E\subset [ 0,2\pi )$ that has linear measure zero, such
that if $\psi _{0}\in [ 0,2\pi )-E$, then there is a
constant $R_{0}=R_{0}(\psi _{0}) >0$ such that for all
$z$ satisfying $\arg z=\psi _{0}$ and $|z| \geq R_{0}$
and for all $(k,j) \in \Gamma $, we have
$$
\Big| \frac{w^{(k) }(z) }{w^{(j)}(z) }\Big| \leq | z| ^{(k-j) (\rho -1+\varepsilon ) }.
$$
\end{lemma}


\subsection*{Proof of Theorem \ref{thm1}}

Suppose that $f\not\equiv 0 $ is a solution of (1.1) with $\rho (f) <\infty $.
Set $\delta =\rho (f)$. Then from Lemma 1, there exists a real
constant $\psi_{0}$ where $\theta _{1}\leq \psi _{0}\leq \theta _{2}$, such
that
\begin{equation}
\big| \frac{f^{(k) }(z) }{f(z) }\big|=o(1) | z| ^{k\,\delta },\quad
k=1,\dots,n \tag{2.5} % 3.1
\end{equation}
as $z\to \infty $ with $\arg z=\psi _{0}$. Then from (2.5) and (1.1),
we obtain that
\begin{equation}
| A_{0}(z) | \leq o(1) | z|^{\delta }| A_{1}(z) | +\dots+o(1) |
z| ^{(n-1) \,\delta }| A_{n-1}(z) |+o(1) | z| ^{n\,\delta } \tag{2.6} %3.2
\end{equation}
as $z\to \infty$ with $\arg z=\psi _{0}$. However this contradicts (2.1) and
(2.2). Therefore, every solution $f\not\equiv 0$ of (1.1) has infinite order.

Next we give an example that illustrates Theorem \ref{thm1}.

\paragraph{Example 1.} Consider the differential equation
\begin{equation}
f''-(3+6e^{z}) f''+(2+6e^{z}+11e^{2z}) f'-6e^{3z}f=0  \tag{2.7}
\end{equation}
In this equation,  for $z=re^{i\theta}$, $r\to +\infty $,
$\frac{\pi }{6}\leq \theta \leq \frac{\pi }{4}$ we have
\begin{gather*}
| A_{0}(z) | =\ | -6\,e^{3z}|
=6e^{3r\cos \theta }>e^{3\frac{\sqrt{2}}{2}r}, \\
| A_{1}(z) | = 2+6e^{z}+11e^{2z}| \leq
19e^{2r\cos \theta }\leq 19e^{\sqrt{3}\,r}<e^{2r}\,\\
| A_{2}(z) | =| -(3+6e^{z}) |
\leq 9e^{r\cos \theta }\leq 9e^{\frac{\sqrt{3}}{2}r}<e^{2r}.
\end{gather*}
As we see, conditions (2.1) and (2.2) of Theorem \ref{thm1} are verified.
The three linearly independent functions
$f_{1}(z) =e^{e^{z}}$, $f_{2}(z) =e^{2e^{z}}$, $f_{3}(z) =e^{3e^{z}}$
are solutions of (2.7) with $\rho(f_{1})=\rho(f_{2})=\rho(f_{3}) =\infty$.

Next we give a generalization of Example 1.

\paragraph{Example 2.} Consider the differential equation
\begin{equation}
f^{^{(n) }}+P_{n-1}(e^{z}) f^{^{(n-1)}}+\dots+P_{1}(e^{z}) f'
+\beta e^{\alpha z}f=0\,, \tag{2.8} %4.4}
\end{equation}
where $\alpha \in \mathbb{R}$, $\alpha >0$, $\beta \in \mathbb{C}$,
$| \beta | \geq 1$, and $P_{1},\dots, P_{n-1}$
are polynomials. If we take the sector
$\theta _{1}\leq \arg z\leq \theta_{2}$, $\theta _{1}$,
$\theta _{2}\in ] 0,\frac{\pi }{2}[ $ with $\theta _{1}$ near enough to
$\theta _{2}$ such that
$\max_{1\leq k\leq n-1} \deg (P_{k}) <\alpha \frac{\cos \theta _{2}}
{\cos\theta _{1}}$, then conditions (2.1) and (2.2) of
Theorem \ref{thm1} are satisfied as $z\to\infty $ with
$\theta _{1}\leq \arg z\leq \theta _{2}$.
 From Theorem \ref{thm1}, it follows that every solution $f\not\equiv 0$
of (2.8) has infinite order.

\subsection*{Proof of Theorem \ref{thm2}}

Suppose that $f\not\equiv 0$ is a solution of (1.1) where $\rho (f) <\infty $
and we set $\beta =\rho (f)$. From Lemma 1, there exists a set
$E\subset [ 0,2\pi )$ that has linear measure zero, such that if
$\psi _{0}\in [ \Phi _{k},\theta _{k})-E$ for some $k$,  then
\begin{equation}
| \frac{f^{(l) }(z) }{f(z) }|=O(| z| ^{l\beta }) ,\quad l=1,\dots,n  \tag{2.9} %5.1
\end{equation}
as $z\to \infty $ with $\arg z=\psi _{0}$. From (2.9) ,(2.4) and (1.1),
we obtain that
\begin{equation}
| A_{0}(z) | \leq | \frac{f^{(n) }}{f}| +| A_{n-1}(z) | | \frac{f^{(
n-1) }}{f}| +\dots+| A_{1\,}(z) |
|\frac{f^{^{/}}}{f}| =O(| z| ^{\sigma }) \tag{2.10} %5.2}
\end{equation}
as $z\to \infty $ with $\arg z=\psi _{0}$, where $\sigma=\alpha +n\beta $.
Let $\varepsilon >0$ be a small constant that satisfies
$\rho (A_{0}) <\frac{\pi }{\mu +2\varepsilon }$
(this is possible since $\rho (A_{0}) <\frac{\pi }{\mu }$).
By using the Phragm\'{e}n-Lindel\"of theorem on (2.10), it can be deduced
that for some integer $s>0$
\begin{equation}
| A_{0}(z) | =O(| z| ^{s}) \tag{2.11}  %5.3}
\end{equation}
as $z\to \infty $ with $\Phi _{k}+\varepsilon \leq \arg z\leq
\theta _{k}-\varepsilon $ for $k=1,\dots,m$.

Now for each $k$, we have from (2.3) that
$\Phi _{k+1}+\varepsilon -(\theta _{k}-\varepsilon ) \leq \mu
+2\varepsilon $, and  so $\rho (A_{0}) <\frac{\pi }{\Phi
_{k+1}-\theta _{k}+2\varepsilon }$. Hence using the Phragm\'{e}n-Lindel\"of
 theorem on (2.11) we can deduce that $|A_{0}(z) |=O(| z|^{s})$
as $z\to \infty $ in the whole complex plane.
This means that $A_{0}(z)$ is a polynomial which contradicts
our hypothesis and completes the proof of Theorem \ref{thm2}.

 Next we give an example that illustrates Theorem \ref{thm2}.

\paragraph{Example 3.} If $A_{0}(z)$ is transcendental with
$\rho (A_{0}) <2$, then from Theorem \ref{thm2}, every solution $f\not\equiv 0$
of the equation
$$
f^{(n) }+P_{n-1}(z) f^{(n-1)}+\dots+P_{2}(z) f^{^{\prime \prime }}
+(e^{z^{3}}+e^{i\,z^{3}}) f'+A_{0}(z) f=0\,, %\tag{5.4}
$$
where $P_{n-1},\dots,P_{2}$ are polynomials, is of infinite order.

\paragraph{Acknowledgement.} The authors would like to thank
the referee for his/her helpful remarks and suggestions.

\begin{thebibliography}{0} \frenchspacing

\bibitem{g1} G. Gundersen, \textit{Finite order
solutions of second order linear differential equations}, Trans. Amer. Math.
Soc. 305 (1988), pp. 415-429.

\bibitem{g2} G. Gundersen, \textit{Estimates for the
logarithmic derivative of a meromorphic functions, plus similar estimates},
 J. London Math. Soc. (2) 37 (1988), pp. 88-104.

\bibitem{g3} G. Gundersen, M. Steinbart M., and S. Wang,
\textit{The possible orders of solutions of linear differential
equations with polynomial coefficients}, Trans. Amer. Math. Soc. 350
(1998), pp. 1225-1247.

\bibitem{h1} W. K. Hayman W. K, \textit{Meromorphic functions},
Clarendon Press, Oxford, 1964.

\bibitem{j1} G. Jank and L. Volkmann, \textit{Einf\"uhrung in die
Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf
Differentialgleichungen},  Birkh\"auser, Basel-Boston-Stuttgart, 1985.

\bibitem{v1} G. Valiron, \textit{Lectures on the general theory of
integral functions}, translated by E. F. Collingwood, Chelsea,
New York, 1949.

\bibitem{w1} H. Wittich,  \textit{Neuere Untersuchungen
\"uber eindeutige analytishe Funktionen},  2nd Edition, Springer-Verlag,
Berlin-Heidelberg-New York, 1968.

\end{thebibliography}

\noindent\textsc{Benharrat Bela\"idi}
(e-mail:belaidi.benharrat@caramail.com)\\
\textsc{Saada Hamouda } (e-mail: hamouda.saada@caramail.com)\\[3pt]
Department of Mathematics, University of Mostaganem \\
B. P.  227 Mostaganem,  Algeria

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