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\markboth{\hfil Order and hyper-order of entire solutions \hfil EJDE--2003/17}
{EJDE--2003/17\hfil Benharrat Bela\"\i di \&  Karima Hamani \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2003}(2003), No. 17, pp. 1--12. \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} \\
 %
  Order and hyper-order of entire solutions of linear
  differential equations with entire coefficients
 %
\thanks{ {\em Mathematics Subject Classifications:} 30D35, 34M10, 34C11.
\hfil\break\indent
{\em Key words:} Linear differential equations, growth of entire functions, 
  hyper-order.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Submitted November 27, 2002. Published February 20, 2003.} }
\date{}
%
\author{Benharrat Bela\"\i di \&  Karima Hamani}
\maketitle

\begin{abstract}
 In this paper, we investigate the growth of solutions of
 the differential equation
 $$
 f^{(k)}+A_{k-1}(z)f^{( k-1) }+\dots+A_{1}(z) f'+A_{0}(z) f=0,
 $$
 where $A_{0}(z) ,\dots, A_{k-1}(z)$
 are entire functions with $A_{0}(z) \not\equiv 0$.
 We will show that if the coefficients satisfy certain
 growth conditions, then every finite order solution of the equation will
 satisfy certain other growth conditions. We will also find conditions on the
 coefficients so that every solution  $f\not\equiv 0$ will have infinite order
 and we estimate in one case the lower bounds of the hyper-order.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}

\section{Introduction and statement of results}

We use the standard notations of the Nevanlinna theory \cite{h1}.
The order of an entire function $f$ is defined as
\[
\ \sigma ( f) =\limsup_{r\to \infty }\frac{\log T(r,f)}{\log r}
=\limsup_{r\to +\infty}\frac{\text{log log }M( r,f)}{\log r},  \tag{1.1}
\]
where $T( r,f)$ is the Nevanlinna characteristic
function of $f$  and $M(r,f) =\max_{| z| =r}| f(z) |$ (see \cite{h1}).
To express the rate of growth of entire function of
infinite order, we recall the following concept.

\paragraph{Definition}
The hyper-order \cite{b1,k1} of an entire function $f$ is defined as
\[
\sigma _{2}( f) =\limsup_{r\to \infty } \frac{\log \log T( r,f) }{\log r}
=\limsup_{r\to \infty } \frac{\log \log \log M( r,f) }{\log r}.  \tag{1.2}
\]
 We define the linear measure of a set $E\subset [ 0,+\infty
) $ by $m( E) =\int_{0}^{+\infty }\chi _{E}( t)
dt$ where $\chi _{E}$ is the characteristic function of $E$. The upper and
the lower densities of $E$ are defined by
\[
\overline{\mathop{\rm dens}}E=\limsup_{r\to +\infty }
\frac{m( E\cap [ 0,r] ) }{r},
\quad \underline{\mathop{\rm dens}}E=\liminf_{r\to +\infty }
\frac{m( E\cap [0,r] ) }{r}.  \tag{1.3}
\]
For $k\geq 2$, we consider the linear differential equation
\[
f^{( k) }+A_{k-1}(z) f^{( k-1)}+\dots+A_{1}(z) f'+A_{0}(z) f=0,
\tag{1.4}
\]
where $A_{0}(z) ,\dots$, $A_{k-1}(z) $ are entire
functions with  $A_{0}(z) \not\equiv 0$.
It is well-known that all  solutions of  (1.4) are
entire functions, and if some coefficients of (1.4)
are transcendental, then (1.4) has at least one solution
with order $\sigma( f) =+\infty $.
A question arises:
\begin{quote}
What conditions on $A_{0}(z) ,\dots,A_{k-1}(z) $
 will guarantee that every solution $f \not\equiv 0$, of (1.4) has
infinite order?
\end{quote}
According to \cite{g3}, \cite[pp. 106-108]{v1} and \cite[pp. 65-67]{w1}, if
$A_{0}(z) ,\dots,A_{k-1}(z) $ are
polynomials with $A_{0}(z)\not\equiv 0$,  then every solution of
(1.4) is an entire function with finite rational order.

In this paper we prove results concerning the above question and other
results about solutions of finite order and their properties.

In the study of the differential equation
\[
f''+A(z) f'+B(z) f=0,
\tag{1.5}
\]
where $A(z)$ and $B(z)\not\equiv 0$ are entire functions,
Gundersen proved the following result.

\begin{theorem}[{\cite[p. 417]{g1}}] \label{thmA}
Let $A(z)$  and $B(z)\not\equiv 0$
be entire functions such that for real constants $\alpha$, $\beta$,
$\theta _{1}$, $\theta_{2}$ satisfy $\alpha >0$, $\beta >0$, and
$\theta _{1}<\theta _{2}$, we have
\[
| A(z) | \geq\exp \{( 1+o(1) ) \alpha | z| ^{\beta }\}\tag{1.6}
\]
and
\[
| B(z) | \leq\exp \{ o( 1)| z| ^{\beta }\}  \tag{1.7}
\]
as $z\to \infty $ in $\theta _{1}\leq \arg z\leq \theta _{2}$.
Let $\varepsilon >0$  be a given small constant, and let
$S( \varepsilon )$ denote the angle
$\theta _{1}+\varepsilon \leq \arg z\leq \theta _{2}-\varepsilon $.
If $f\not\equiv 0$ is a solution of (1.5),
where $\sigma ( f) <+\infty $, then the following conditions hold:
\begin{itemize}
\item[(i)] There exists a constant $b\neq 0$ such that $f(z) \to b$
 as $z\to \infty $ in $S( \varepsilon ) $.
Furthermore, as $z\to \infty $ in $S(\varepsilon)$,
\[
| f(z) -b| \leq \exp \{-(1+o( 1) ) \alpha | z| ^{\beta }\}
\tag{1.8}
\]
\item[(ii)] For each integer $k\geq 1$, as $z\to \infty $
in $S( \varepsilon )$,
\[
| f^{( k) }(z) |\leq \exp\{ -\,( 1+o( 1) ) \alpha | z| ^{\beta}\}.  \tag{1.9}
\]
\end{itemize}
\end{theorem}

In the same paper, Gundersen proved the following statement.

\begin{theorem}[{\cite[p. 418]{g1}}] \label{thmB}
Let $A(z)$  and $B(z)\not\equiv 0$ be entire functions, and let
$\alpha >0$, $\beta >0$ be constants, with $\sigma(B) <\beta$.
Suppose that for any given $\varepsilon >0$,
there exist two finite collections of real numbers
$\{\phi_{k}\} $ and $\{\theta_{k}\}$ that satisfy
$\phi _{1}<\theta _{1}<\phi _{2}<\theta _{2}<\dots<\phi_{n}<\theta _{n}
<\phi _{n+1}$,  where $\phi _{n+1}=\phi _{1}+2\pi $ and
\[
\sum_{k=1}^n ( \phi _{k+1}-\theta _{k}) <\varepsilon ,  \tag{1.10}
\]
such that
\[
| A(z) | \geq \exp \{ ( 1+o( 1)
) \alpha | z| ^{\beta }\}  \tag{1.11}
\]
as $z\to \infty $ in $\phi _{k}\leq \arg z\leq \theta _{k}$ $( k=1,2,\dots n)$.
Then every solution $f\not\equiv 0$  of (1.5) is of infinite order.
\end{theorem}

Recently, for the second order equation (1.5),
Ki-Ho Kwon obtained in \cite{k1} the following results:

\begin{theorem}[{\cite[p. 488]{k1}}] \label{thmC}
Let $A(z) $ and $B(z)$ be entire functions satisfying
$\sigma ( B) <\sigma(A)$. Then every solution $f\not\equiv 0$
of finite order of (1.5) satisfies $\sigma (f) \geq \sigma ( A)$.
\end{theorem}

\begin{theorem}[{\cite[p. 489]{k1}}] \label{thmD}
Let $A(z) $ and $B(z)$ be entire functions where
$0<\sigma (B) <1/2$, and let there exist a real constant
$\beta <\sigma (B) $ and a set $E_{\beta }\subset [0,+\infty )$
with $\underline{\mathop{\rm dens}}E_{\beta }=1$
such that for all $r\in E_{\beta }$, we have
\[
\min_{|z|=r} |A(z)| \leq \exp ( r^{\beta }) .  \tag{1.12}
\]
Then every solution $f\not\equiv 0$ of (1.5) is of infinite order with
\[
\sigma _{2}( f) =\limsup_{r\to \infty }\frac{\log \log T( r,f) }{\log r}
\geq \sigma ( B) .
\]
\end{theorem}

We shall investigate generalizations of problems of the
above type to higher order homogeneous linear differential equations.

\begin{theorem} \label{thm1}
Let $A_{0}(z),\dots,A_{k-1}(z)$ with $A_{0}(z) \not\equiv 0$
be entire functions such that for real constants $\alpha $, $\beta $,
$\theta _{1}$, $\theta _{2}$,  where $\alpha >0$, $\beta >0$, and
$\theta_{1}<\theta _{2}$, we have
\[
| A_{1}(z) | \geq\exp \{ (1+o( 1) ) \alpha | z| ^{\beta }\}\tag{1.13}
\]
and
\[
| A_{j}(z) | \leq\exp \{ o(1) | z| ^{\beta }\}, \quad  j=0,2,\dots,k-1
\tag{1.14}
\]
as $z\to \infty $ in $\theta _{1}\leq \arg z\leq \theta _{2}$.
Let $\varepsilon >0$  be a given small
constant, and let $S( \varepsilon ) $  denote the angle
$\theta _{1}+\varepsilon \leq \arg z\leq \theta _{2}-\varepsilon $.

If $f\not\equiv 0$ is a solution of (1.4)
 where $\sigma ( f) <+\infty $,  then the following conditions hold:
\begin{itemize}
\item[(i)] There exists a constant $b\neq 0$
such that $f(z) \to b$  as $z\to \infty $  in $S( \varepsilon ) $.
Furthermore, as $z\to \infty$  in $S( \varepsilon )$,
\[
| f(z) -b| \leq \exp \{ -\,( 1+o(
1) ) \alpha | z| ^{\beta }\}.  \tag{1.15}
\]
\item[(ii)]  For each integer $m\geq 1$, as $z\to \infty $ in $S( \varepsilon )$,
\[
| f^{( m)}(z)|\leq \exp \{-\,( 1+o( 1) ) \alpha | z| ^{\beta}\}.\tag{1.16}
\]
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm2}
Let $A_{0}(z) ,\dots,A_{k-1}(z)$  with $A_{0}(z)\not\equiv 0$
be entire functions, and let $\alpha >0$, $\beta >0$  be constants where
$\sigma (A_{j}) <\beta $ ($j=0,2,\dots,k-1$).
Suppose that for any given $\varepsilon >0$,  there exist two finite
collections of real numbers $\{\phi _{s}\}$ and $\{ \theta_{s}\}$
that satisfy
$\phi _{1}<\theta _{1}<\phi _{2}<\theta _{2}<\dots<\phi _{n}<\theta _{n}
<\phi _{n+1}$ where $\phi _{n+1}=\phi _{1}+2\pi $ and
\[
\sum _{s=1}^n (\phi _{s+1}-\theta _{s})<\varepsilon ,  \tag{1.17}
\]
such that
\[
| A_{1}(z) | \geq \exp \{ ( 1+o(1) ) \alpha | z| ^{\beta }\}  \tag{1.18}
\]
as $z\to \infty $  in $\phi _{s}\leq \arg z\leq \theta _{s}$
($s=1,2,\dots n$). Then every solution
$f\not\equiv 0$ of (1.4) is of infinite order.
\end{theorem}

Here we estimate the lower bounds for the order of solution
$f\not\equiv 0$ of (1.4) when $f$ is of finite order.

\begin{theorem} \label{thm3}
Let $A_{0}(z),\dots,A_{k-1}(z)$ be entire functions that satisfy \\
$\max \{ \sigma ( A_{j}) : j=0,2,\dots,k-1\} <\sigma ( A_{1}) $.
Then every solution  $f\not\equiv 0$ of (1.4) of finite order satisfies
$\sigma ( f) \geq\sigma ( A_{1}) $.
\end{theorem}

\paragraph{Remark} If $\max\{\sigma ( A_{j}): j=0,2,\dots,k-1\}=\sigma ( A_{1})$,
the conclusion of Theorem \ref{thm3} is in general false. Indeed $f(z) =z^{2}$
satisfies $f'' +z^{2}e^{z}f''+ze^{z}f'-4e^{z}f=0$.

Now, we estimate the lower bounds on the hyper-order if every
solution  $f\not\equiv 0$ is of infinite order.

\begin{theorem} \label{thm4}
Let $A_{0}(z) ,\dots,A_{k-1}(z)$ be entire functions where
$0<\sigma ( A_{0}) <1/2$, and let there exist a real
constant $\beta <\sigma ( A_{0}) $ and a set
$E_{\beta}\subset [ 0,+\infty ) $ with
$\underline{\mathop{\rm dens}}E_{\beta }=1$ such that for all
$r\in E_{\beta }$, we have
\[
\min_{| z| =r}| A_{j}(z) | \leq
\exp ( r^{\beta }) \quad  j=1,2,\dots,k-1 . \tag{1.19}
\]
Then every solution $f\not\equiv 0$ of (1.4) is of infinite order with
\[
\sigma _{2}( f) =\limsup_{r\to \infty }
\frac{\log \log T( r,f) }{\log r}\geq \sigma ( A_{0}) .
\]
\end{theorem}

\section{Preliminary Lemmas}

We need the following lemmas for the proofs of our theorems.

\begin{lemma}[{\cite[p. 89]{g2}}] \label{lm1}
Let $w$ be a transcendental entire function of
finite order $\sigma $. Let
$\Gamma =\{(k_{1},j_{1}) ,( k_{2},j_{2}) ,\dots,( k_{m},j_{m})\}$
 denote a finite set of distinct pairs of integers
that satisfy  $k_{i}>j_{i}\geq 0$, $i=1,\dots,m,$ 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
\[
\Big| \frac{w^{( k) }(z) }{w^{( j)
}(z) }\Big| \leq | z| ^{( k-j) (
\sigma -1+\varepsilon ) }.  \tag{2.1}
\]
\end{lemma}


\begin{lemma}[{\cite[p. 421]{g1}}] \label{lm2}
Let $w$ be an entire function such that $| w'(z) | $ is unbounded on some
ray $\arg z=\theta$. Then there exists
an infinite sequence of points $z_{n}=r_{n}e^{i\,\theta}$ where
$r_{n}\to+\infty $ such that $w'( z_{n})\to \infty $ and
\[
\big| \frac{w( z_{n}) }{w'( z_{n}) }\big| \leq ( 1+o( 1) ) | z_{n}| ,
\tag{2.2}
\]
as $z_{n}\to \infty $.
\end{lemma}

\begin{lemma}[{\cite[p. 421]{g1}}] \label{lm3}
Let $w$ be analytic on a ray $\arg z=\theta $, and suppose that for some
constant $\alpha >1$, we have
\[
\big| \frac{w'(z) }{w(z) }\big|=O( | z| ^{-\alpha }) ,  \tag{2.3}
\]
as $z\to \infty $ along $\arg z=\theta $.
Then there exists a constant $c\neq 0$ such that $w(z)\to c$ as
$z\to \infty $ along $\arg z=\theta $.
\end{lemma}

\begin{lemma}[\cite{b2}] \label{lm4}
Let $f(z) $ be an entire function of order $\sigma $ where
$0<\sigma <1/2$, and let $\varepsilon >0$
be a given constant. Then there exists a set
$E_{2}\subset [0,+\infty )$ with $\overline{\mathop{\rm dens}} E_{2}\geq
1-2\sigma $  such that for all $z$ satisfying $|z| =r\in E_{2}$, we have
\[
| f(z) | \geq \exp ( r^{\sigma -\varepsilon}) .  \tag{2.4}
\]
\end{lemma}

\begin{lemma}[\cite{b1}] \label{lm5}
Let $H$ be a set of complex numbers satisfying
$\overline{\mathop{\rm dens}}\{ | z| :z\in H\} >0$, and let
$A_{0}(z) ,\dots,A_{k-1}(z) $ be entire functions such that for some
constants $0\leq \beta <\alpha $ and $\mu >0$, we have
\[
| A_{0}(z) | \geq \exp ( \alpha | z| ^{\mu })  \tag{2.5}
\]
and
\[
| A_{j}(z) | \leq \exp ( \beta | z|^{\mu }) \quad j=1,\dots,k-1  \tag{2.6}
\]
as $\ z\to \infty $ for $z\in H$. Then
every solution  $f\not\equiv 0$ of (1.4) satisfies $\sigma ( f) =+\infty $
and $\sigma _{2}( f) \geq \mu $.
\end{lemma}

\section{Proofs of Theorem \ref{thm1}}

Suppose that $f\not\equiv 0$ is a solution of (1.4)
with $\sigma (f)<\infty $. Set $\delta =\sigma (f)$. Then from Lemma \ref{lm1},
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
\[
\big| \frac{f^{(j)}(z)}{f'(z)}\big| =o(1)|z|^{(j-1) \delta }\quad
 j=2,\dots ,k  \tag{3.1}
\]
as $z\to \infty $ along $\arg z=\psi _{0}$.

Now suppose that $| f'(z) | $ is
unbounded on some ray $\arg z=\phi _{0}$ where
$\phi _{0}\in [ \theta_{1},\theta _{2}] -E$.  Then from Lemma \ref{lm2}, there exists
an infinite sequence of points $z_{n}=r_{n}e^{i\,\phi _{0}}$ where
$r_{n}\to +\infty $ such that $f'( z_{n})\to \infty $ and
\[
\big| \frac{f( z_{n}) }{f'( z_{n}) }\big| \leq ( 1+o( 1) ) | z_{n}| .
\tag{3.2}
\]
From (1.4), we have
\[
| A_{1}(z) | \leq \big| \frac{f^{( k) }}{f'}\big| +| A_{k-1}(z) |
\big| \frac{f^{( k-1) }}{f'}\big| +\dots+| A_{2}(z) |
\big| \frac{f''}{f'}\big| +| A_{0}(z) | \big| \frac{f}{f'}\big| .
\tag{3.3}
\]
By using (3.2), (3.1), (1.13), and (1.14), we will obtain a contradiction
in (3.3) as $z\to \infty $. Therefore, $|f'(z) | $ is bounded on any
ray $\arg z=\phi $ where $\phi \in [ \theta _{1},\theta _{2}] -E$.
It then follows from the classical Phragm\'{e}n-Lindel\"{o}f theorem
\cite[p. 214]{m1}  that there exists a constant $M>0$ such that
\[
| f'(z) | \leq M  \tag{3.4}
\]
for all $z\in S( \varepsilon )$.
If $\theta _{0}\in [ \theta _{1}+\varepsilon ,\theta
_{2}-\varepsilon ] -E$, then when $\arg z=\theta _{0}$, we obtain from
(3.4) that
\[
| f(z) | \leq | f( 0) | +\big|\int_0^z f'( u) du\big| \leq
| f( 0) | +M| z| .  \tag{3.5}
\]
From (3.5), (3.1), and (1.4), we obtain
\[\begin{aligned}
|A_{1}(z)|| f'(z) |
&\leq \Big\{ o(1)|z|^{( k-1) \delta }+o(1)|z|^{(k-2)\delta
}|A_{k-1}(z)|+\dots \\
&\quad+o(1)|z|^{\delta }|A_{2}(z)|\Big\} | f'(z) | +
| A_{0}(z) | \{ | f( 0) | +M| z| \}
\end{aligned}\tag{3.6}
\]
as  $z\to \infty $ along $\arg z=\theta _{0}$. From
(3.6), (1.13), and (1.14), we can deduce that
\[
| f'(z) | \leq \exp \{ -(1+o( 1) ) \alpha | z| ^{\beta }\}
\tag{3.7}
\]
as  $z\to \infty $ along $\arg z=\theta _{0}$. By using an
application of the Phragm\'{e}n-Lindel\"{o}f theorem to (3.7),
 it can be deduced that
\[
| f'(z) | \leq \exp \{ -(1+o( 1) ) \alpha | z| ^{\beta }\}
\tag{3.8}
\]
as $z\to \infty $ in $S( 2\varepsilon )$. This gives $m=1 $ in (1.16).

Now let $z\in S( 3\varepsilon ) $ where $| z| >1$, let
$\gamma $ be a circle of radius one with center at $z$, and let $m\geq 1$ be
an integer. Then from the Cauchy integral formula and (3.8),
we obtain as $z\to \infty $ in $S( 3\varepsilon )$,
\[
\big| f^{( m) }(z) \big| \leq \frac{(
m-1) !}{2\pi }\oint_\gamma \frac{| f'( u) | }{| z-u| ^{m}}| du| \leq
\exp \{ -( 1+o( 1) ) \alpha | z|
^{\beta }\} .  \tag{3.9}
\]
This proves (1.16).
Now fix $\theta $, where $\theta _{1}+\varepsilon \leq \theta \leq \theta
_{2}-\varepsilon $, and set
\[
a_{0}=\int_0^{+\infty } f'(te^{i\theta }) e^{i\theta }dt,  \tag{3.10}
\]
where we note that  $a_{0}\in \mathbb{C}$ from (1.16). Let
$z=| z| e^{i\,\psi }$ where $\theta _{1}+\varepsilon \leq \psi
\leq \theta _{2}-\varepsilon $. Then from the Cauchy theorem and
(3.10), we obtain
\[ \begin{aligned}
f(z) -f( 0) -a_{0}&=\overset{z}{\underset{0}{\int }}
f'( u) du-\int_0^{+\infty} f'( te^{i\theta }) e^{i\theta }dt\\
&=\int_\theta^\psi f'( |z| e^{i\zeta }) i| z| e^{i\zeta }d\zeta
-\int_{| z| }^{+\infty} f'(te^{i\theta }) e^{i\theta }dt.
\end{aligned} \tag{3.11}
\]
Since
\[
| f'(z) | \leq \exp \{ -(1+o( 1) ) \alpha | z| ^{\beta }\}
\]
as $z\to \infty $ in $S( \varepsilon ) $, then from
(3.11), we get
\[\begin{aligned}
&| f(z) -b|\\
&=| \int_\theta^\psi f'( | z| e^{i\zeta }) i|z| e^{i\zeta }d\zeta
-\int_{| z|}^{+\infty} f'( te^{i\theta }) e^{i\theta }dt|\\
&\leq | \psi -\theta | | z| \exp \{ -(1+o( 1) ) \alpha | z| ^{\beta }\}
+\int_{| z| }^{+\infty } \exp \{ -(1+o( 1) ) \alpha t^{\beta }\} dt\\
&\leq | \psi -\theta | | z| \exp \{ -(1+o( 1) ) \alpha | z| ^{\beta }\} \\
&\quad+\frac{1}{( 1+o( 1) ) \alpha \beta \frac{|z| ^{\beta -1}}{2}
\exp \{ ( 1+o( 1) ) \alpha \frac{| z| ^{\beta }}{2}\} }
\int_{| z| }^{+\infty } \frac{( 1+o( 1) )
\alpha \beta \frac{t^{\beta -1}}{2}}{\exp \{ ( 1+o( 1)) \alpha
 \frac{t^{\beta }}{2}\} }dt\\
&\leq | \psi -\theta | | z| \exp \{ -(1+o( 1) ) \alpha | z| ^{\beta }\} \\
&\quad+\frac{1}{( 1+o( 1) ) \alpha \beta \frac{|
z| ^{\beta -1}}{2}\exp \{ ( 1+o( 1) )
\alpha \frac{| z| ^{\beta }}{2}\} }\exp \{ -(1+o( 1) ) \alpha
\frac{| z| ^{\beta }}{2}\} \\
&\leq \exp \{ -( 1+o( 1) ) \alpha | z|^{\beta }\}
\end{aligned}\tag{3.12}
\]
as $z\to \infty $ in $S( \varepsilon ) $, where $b=f( 0) +a_{0}$.
Note that $a_{0}$ in (3.10) is independent of $\theta$.
Since (3.12) is the inequality (1.15), it remains only to show that $b\neq 0$.

From Lemma \ref{lm1}, there exists a ray $\arg z=\psi _{1}$ where
$\theta_{1}+\varepsilon \leq \psi _{1}\leq \theta _{2}-\varepsilon $,
 such that
\[
| \frac{f^{(j)}(z)}{f(z)}| =o(1)|z|^{j\,\delta }\quad
j=2,\dots ,k  \tag{3.13}
\]
as  $z\to \infty $ along $\arg z=\psi _{1}$, where $\delta =\sigma (f)$.
Then from  (3.13), (1.13), (1.14), and (1.4), we have
\[\begin{aligned}
| \frac{f'(z) }{f(z) }|
&\leq | \frac{A_{0}(z) }{A_{1}(z) }| +|A_{2}(z) | | \frac{f^{''}(
z) }{f(z) A_{1}(z) }| +\dots \\
&\quad +| A_{k-1}(z) | | \frac{f^{^{( k-1)
}}(z) }{f(z) A_{1}(z) }| +|\frac{f^{^{( k) }}(z) }{f(z) A_{1}(
z) }|\\
&\leq \exp \{ -( 1+o( 1) )
\alpha | z| ^{\beta }\}
\end{aligned}\tag{3.14}
\]
as  $z\to \infty $ along $\arg z=\psi _{1}$. By applying
Lemma \ref{lm3} to (3.14), and noting that $f(z)\to b$ as
$z\to \infty $ in $S( \varepsilon
) $ from (3.12), we see that $b\neq 0$. The proof of
Theorem \ref{thm1} is complete. \hfill$\diamondsuit$ \smallskip

The next example illustrates Theorem \ref{thm1}.


\begin{example} \label{ex3.1} \rm
Consider the differential equation
\[
f'''+e^{z}f''-2e^{-z}f'+( 1-e^{z}) f=0.  \tag{3.15}
\]
In this equation, for $z=re^{i\theta }( r\to +\infty )$ and
$\frac{2\pi }{3}\leq \theta \leq \frac{3\pi }{4}$, we have
\begin{gather*}
|A_{1}(z)|=\ |-2e^{-z}|=2\exp ( -r\cos \theta )
\geq \exp (( 1+o( 1) ) \frac{r}{2}) , \\
|A_{0}(z)|=| 1-e^{z}| \leq 1+\exp ( r\cos \theta )
\leq \exp ( o( 1) r) \, \\
|A_{2}(z)|=|e^{z}|=\exp ( r\cos \theta ) \leq \exp ( o(1) r) .
\end{gather*}
It is easy to see that the conditions (1.13) and (1.14) of  Theorem \ref{thm1}
 are satisfied ($\alpha =\frac{1}{2}$, $\beta=1)$. The function $f(z)=e^{z}+2$
with $\sigma ( f) =1$ satisfies equation (3.15) and the relations (1.15),
(1.16).
\end{example}

\section{Proof of Theorem \ref{thm2}}

Suppose that $f\not\equiv 0$ is a solution of (1.4)
where $\sigma (f)<\infty $. Let $\varepsilon >0$, $\{ \phi _{s}\}
$ and $\{ \theta _{s}\} $ be as in the
hypothesis. From (1.18) and $\sigma ( A_{j})<\beta $ ($j=0,2,\dots,k-1$),
it follows from Theorem \ref{thm1} (i) that $| f(z) | $ is bounded within each
angle $\phi _{s}+\varepsilon \leq \arg z\leq \theta _{s}-\varepsilon $
 ($s=1,2,\dots n$). Since $\varepsilon $ can be
arbitrarily small, it follows from (1.17) and the
Phragm\'{e}n-Lindel\"{o}f theorem that $|f(z)|$ is bounded in the whole
finite plane. Thus $f$ is a nonzero constant from Liouville's theorem, and
this contradicts (1.4).

The next two examples illustrate Theorem \ref{thm2}.



\begin{example} \label{ex4.1} \rm
Consider the differential equation
\[
\text{\ }f''' +\cos \sqrt{z}f''+\sin z\text{ }f'
+\frac{\sin \sqrt{z}}{\sqrt{z}}f=0.  \tag{4.1}
\]
Then from Theorem \ref{thm2}, it follows that every solution $f\not\equiv 0$
is of infinite order.
\end{example}

\begin{example} \label{ex4.2} \rm
Suppose $A_{0}(z) ,\dots, A_{k-1}(z)$ with $A_{0}(z)\not\equiv 0$
 be entire functions and $n\geq 1$ an integer. Then from
Theorem \ref{thm2}, it follows that every solution $f\not\equiv 0$ of
following two equations is of infinite order:
\[
f^{( k) }+A_{k-1}(z) f^{( k-1) }+\dots+A_{2}(z) f''+\sin
( z^{n}) f'+A_{0}(z) f=0,  \tag{4.2}
\]
where $\rho ( A_{j}) <n$, $j=0,2,3,\dots,k-1$; and
\[
f^{( k) }+A_{k-1}(z)f^{( k-1) }+\dots+A_{2}(z) f''+\cos( z^{n/2}) f'
+A_{0}(z) f=0,  \tag{4.3}
\]
where $\rho ( A_{j}) <n/2$, $j=0,2,3,\dots,k-1$.
\end{example}

\section{Proof of Theorem \ref{thm3}}

Set $\max \{ \sigma ( A_{j}):j=0,2,\dots,k-1\}=\beta<\alpha= \sigma ( A_{1})$.
 Suppose that $f\not\equiv 0$ is a solution of (1.4) with
 $\sigma (f) <+\infty $. It follows from (1.4), that
\[
-A_{1}(z) =\frac{f^{( k) }}{f'}+A_{k-1}(z) \frac{f^{( k-1) }}{f'}%
+\dots+A_{2}(z) \frac{f''}{f'}+A_{0}(z) \frac{f}{f'}.  \tag{5.1}
\]
Hence from Nevanlinna's fundamental results on meromorphic
functions \cite[p. 5]{h1}, \cite[Theorem 2.2, p. 34]{h1}  and
\cite[Theorem 3.1, p. 55]{h1}, we have
\[ \begin{aligned}
m( r,A_{1}) &\leq m( r,A_{0}) +m( r,A_{2})
+\dots+m( r,A_{k-1}) \\
&\quad+\sum_{j=2}^{k}m( r,\frac{f^{( j) }}{f'})
+m( r,\frac{f}{f'}) +O( 1) \\
&\leq m( r,A_{0}) +m( r,A_{2}) +\dots+m(
r,A_{k-1}) +m( r,\frac{f}{f'}) +O( \log r)
\end{aligned}\tag{5.2}
\]
holds for all $r$ outside a set $E\subset ( 0,+\infty )$ with a
linear measure $m(E) =\delta <+\infty $. Here the notation $m( r,h)$
for a meromorphic function $h$ is defined by
\[
m( r,h) =\frac{1}{2\pi }\int_{0}^{2\pi }\log ^{+}| h(re^{i\theta }) |
 d\theta ,  \tag{5.3}
\]
which is equal to $T(r,h)$ if $h$ is entire. It follows from (5.2) that
for $r\notin E$
\[
T( r,A_{1}) \leq T( r,A_{0}) +T( r,A_{2})
+\dots+T( r,A_{k-1})
+2T( r,f) +O( \log r),
\tag{5.4}
\]
since $m( r,f/f') \leq 2T( r,f)+O( 1)$, $r\notin E$.
Since $\sigma (A_{1}) =\alpha $, there exists
$\{ r_{n}'\} $  with $r_{n}'\to \infty$, such that
\[
\lim_{r_{n}'\to \infty } \frac{\log T(r_{n}',A_{1}) }{\log r_{n}'}=\alpha .
\tag{5.5}
\]
By $m( E) =\delta <+\infty $, there exists a point
$r_{n}\in [ r_{n}',r_{n}'+\delta +1] -E$. From
\[
\frac{\log T( r_{n},A_{1}) }{\log r_{n}}\geq \frac{\log T(
r_{n}',A_{1}) }{\log ( r_{n}'+\delta
+1) }=\frac{\log T( r_{n}',A_{1}) }{\log
r_{n}'+\log ( 1+( \delta +1) /r_{n}') }  \tag{5.6}
\]
we get
\[
\liminf_{r_{n}\to \infty } \frac{\log T(
r_{n},A_{1}) }{\log r_{n}}\geq \alpha .  \tag{5.7}
\]
So, for any given $\varepsilon$ with $0<2\varepsilon <\alpha -\beta$,
and for $j=0,2,\dots,k-1$,
\[
T( r_{n},A_{j}) \leq r_{n}^{\beta +\varepsilon }\quad\text{and}\quad
T( r_{n},A_{1}) >r_{n}^{\alpha -\varepsilon }  \tag{5.8}
\]
hold for sufficiently large $r_{n}$. By (5.4) and (5.8), we get for
sufficiently large $r_{n}$,
\[
r_{n}^{\alpha -\varepsilon }\leq ( k-1) r_{n}^{\beta +\varepsilon
}+2T( r_{n},f) +O( \log r_{n}) .  \tag{5.9}
\]
Therefore,
\[
\limsup_{r_{n}\to \infty } \frac{\log T(r_{n},f) }{\log r_{n}}
\geq \alpha -\varepsilon  \tag{5.10}
\]
and since $\varepsilon $ is arbitrary, we get
$\sigma ( f) \geq \sigma ( A_{1}) =\alpha $. This proves Theorem \ref{thm3}.
\hfill$\diamondsuit$

The next example illustrates Theorem \ref{thm3}.

\begin{example} \label{ex5.1} \rm
The equation
\[
f'''' +e^{-z}f'-16f=0  \tag{5.11}
\]
has a solution $f(z) =120e^{2z}+16e^{z}+1$, where
$\sigma (A_{1}) =1$, $\sigma ( A_{j}) =0$, $j=0,2,3$, and $\sigma ( f) =1$.
\end{example}

\section{Proof of Theorem \ref{thm4}}

Let $\beta <\sigma ( A_{0}) $ and let $f\not\equiv 0$ be a solution of
(1.4). Suppose that $\beta <\alpha <\sigma ( A_{0})$ and that there is a
set $E_{\beta }\subset [ 0,+\infty )$ of lower density 1 satisfying
(1.19). Set
\[
E_{1}=\big\{ z:| z| =r\in E_{\beta }\text{ and }|A_{j}(z) |
=\min_{|z| =r}|A_{j}(z) |, j=1,2,\dots,k-1\big\} .
\tag{6.1}
\]
Then $\underline{\mathop{\rm dens}}\{ | z| :z\in E_{1}\text{ }\} =1$
and
\[
| A_{j}(z) | \leq \exp ( r^{\beta }), \quad
j=1,2,\dots,k-1  \tag{6.2}
\]
for all $z\in E_{1}$. Also, from Lemma \ref{lm4}, there is a set
$E_{2}\subset [0,+\infty )$ of positive upper density such that for all $z$
satisfying $|z| \in E_{2}$, we have
\[
| A_{0}(z) | \geq \exp ( r^{\alpha }) .
\tag{6.3}
\]
Now let $E=\{ z\in E_{1}:| z| \in E_{2}\}$. Then,
with a set $E$ and the number $\alpha $, $A_{0}(z),\dots,A_{k-1}(z) $
satisfy the hypothesis of Lemma \ref{lm5} respectively.
Hence we conclude by Lemma \ref{lm5} that every solution $f\not\equiv 0$ of (1.4)
satisfies $\sigma ( f) =+\infty $ and
\[
\sigma _{2}( f) =\limsup_{r\to \infty }
\frac{\log \log T( r,f) }{\log r}\geq \alpha .  \tag{6.4}
\]
Thus the result of the theorem follows since $\alpha $ is arbitrary.
\hfill$\diamondsuit$

The next example illustrates Theorem \ref{thm4}.


\begin{example}  \label{ex6.1} \rm
Let $P_{1}(z)$, \dots, $P_{k-1}(z) $ be nonconstant polynomials,
and let $h_{1}(z)$, \dots, $h_{k-1}(z)$ be entire functions satisfying
$\sigma ( h_{j}) <\deg P_{j}$ for $j=1,\dots,k-1$.
Let $A_{0}(z) $ be an entire function with $0<\sigma ( A_{0}) <1/2$.
Then, by Theorem \ref{thm4}, every solution $f\not\equiv 0$ of the equation
\[
f^{^{( k) }}+h_{k-1}(z) e^{P_{k-1}(z)
}f^{^{( k-1) }}+\dots+h_{1}(z) e^{P_{1}(z)}f'+A_{0}(z) f=0  \tag{6.5}
\]
is of infinite order with $\sigma _{2}( f) \geq \sigma (A_{0}) $,
because
\[
\underset{| z| =r}{\min }| h_{j}(z)
e^{P_{j}(z) }| \to 0,\quad j=1,\dots,k-1
\]
as $r\to \infty $.
\end{example}

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

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\end{thebibliography}

\noindent\textsc{Benharrat Bela\"\i di}\\
\textsc{Karima Hamani}\\
Department of Mathematics,
Laboratory of Pure and Applied Mathematics \\
University of Mostaganem\\
B. P 227 Mostaganem Algeria\\
e-mail: belaidi@univ-mosta.dz\quad belaidi.benharrat@caramail.com
\end{document}