\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 70, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/70\hfil Growth and oscillation of solutions]
{Growth and oscillation of solutions to linear differential
equations with entire coefficients having the same order}

\author[B. Bela\"idi\hfil EJDE-2009/70\hfilneg]
{Benharrat Bela\"idi}

\address{Benharrat Bela\"idi \newline
Department of Mathematics\\
Laboratory of Pure and Applied Mathematics\\
University of Mostaganem\\
B. P. 227 Mostaganem, Algeria}
\email{belaidi@univ-mosta.dz, belaidibenharrat@yahoo.fr}

\thanks{Submitted February 17, 2009. Published June 1, 2009.}
\subjclass[2000]{34M10, 30D35}
\keywords{Linear differential equation; growth of
 entire function; iterated order}

\begin{abstract}
 In this article, we investigate the growth and
 fixed points 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.
 Some estimates are given for the iterated order and iterated
 exponent of convergence of fixed points of solutions of the
 above equation when most of the coefficients have the same
 order with each other.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction and statement of results}

 For the definition of the iterated order of an entire function, we
use the same definition as in \cite{k1}, \cite[p. 317]{b6},
\cite[p. 129]{l1}. For all $r\in \mathbb{R}$, we define
$\exp _{1}r:=e^{r}$ and $\exp _{p+1}r:=\exp (\exp_{p}r)$,
$p\in \mathbb{N}$. We also define for all $r$ sufficiently large
$\log _{1}r:=\log r$ and
$\log _{p+1}r:=\log (\log _{p}r)$, $p\in \mathbb{N}$.
Moreover, we denote by $\exp _{0}r:=r$, $\log _{0}r:=r$,
$\log_{-1}r:=\exp _{1}r$ and $\exp _{-1}r:=\log _{1}r$.

\begin{definition} \label{def1.1} \rm
Let $f$ be an entire function. Then the
iterated $p$-order $\rho _{p}(f)$ of $f$ is defined by
\begin{equation}
\rho _{p}(f)=\limsup_{r\to +\infty }
\frac{\log _{p}T(r, f)}{\text{log }r}
=\limsup_{r\to +\infty } \frac{\log _{p+1}M(r,
 f)}{\text{log }r},
\label{e1.1}
\end{equation}
where $p$ is an integer, $p\geq 1$, $T(r, f)$ is the Nevanlinna
characteristic function of $f$ and
$M(r,f)=\max_{|z|=r}|f(z)|$;
see \cite{h1,n1}. For $p=1$, this notation is
called order and for $p=2$ hyper-order \cite{y1}.
\end{definition}

\begin{definition}[\cite{b6,l1}] \label{def1.2}\rm
 The finiteness degree of the order of an
entire function $f$ is defined by
\begin{equation}
i(f)=\begin{cases}
0,&\text{for $f$ a polynomial}, \\
\min\{ j\in \mathbb{N}: \rho _{j}(f)<+\infty \},
&\text{for $f$ transcendental for which} \\
&\text{there exists  $j\in \mathbb{N}$ with $\rho _{j}(f)<+\infty$}
\\
+\infty , &\text{when $\rho _{j}(f)=+\infty$ for all $j\in \mathbb{N}$}.
\end{cases}  \label{e1.2}
\end{equation}
\end{definition}

\begin{definition} \label{def1.3}\rm
 Let $f$ be an entire function. Then the
iterated $p$-type of an entire function $f$, with iterated $p$-order
$0<\rho_{p}(f)<\infty $ is defined by
\begin{equation}
\tau _{p}(f)=\limsup_{r\to +\infty}
\frac{\log _{p-1}T(r, f)}{ r^{\rho_{p}(f)}}
=\limsup_{r\to +\infty} \frac{\log _{p}M(r, f)}{ r^{\rho _{p}(f)}}
\quad (1\leq p\text{ an integer}).
\label{e1.3}
\end{equation}
\end{definition}

 For $k\geq 2,$\ we consider the linear differential equation
\begin{equation}
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots+A_{1}(z)f'+A_{0}(z)f=0,
\label{e1.4}
\end{equation}
where $A_{0}(z),\dots, A_{k-1}(z)$ are entire
functions. It is well-known that all solutions of equation
\eqref{e1.4} are entire functions and if some of the coefficients of
\eqref{e1.4} are transcendental, then \eqref{e1.4} has at
least one solution with order $\rho (f)=+\infty $.

 A natural question arises: What conditions on
$A_{0}(z),A_{1}(z),\dots ,A_{k-1}(z)$ will guarantee that every solution
$f\not\equiv 0$ has infinite order?
Also: For solutions of infinite order,
how to express the growth of them explicitly, it is a very important
problem. Partial results have been available since in a paper by
Frei \cite{f1}. Extensive work in recent years has been
concerned with the growth of solutions of complex linear differential
equations. Many results have been obtained for the growth and the
oscillation of solutions of the differential equation \eqref{e1.4};
see e.g. \cite{b2,b3,b4,b5,c1,c2,c3,c4,c5,k1,l1,l2,l3,t1,t2,w1}.
Examples of such results are the following two theorems:

\begin{theorem}[\cite{k1}] \label{thmA}
Let $A_{0}(z),\dots$, $A_{k-1}(z)$
be entire functions such that $i(A_{0})=p$ $(0<p<\infty )$.
 If either $\max \{i(A_{j}): j=1,2,\dots,k-1\}<p$  or
$\max \{\rho _{p}(A_{j}): j=1,2,\dots,k-1\}<\rho _{p}(A_{0})$,
 then every solution $f\not\equiv 0$  of
\eqref{e1.4}  satisfies $i(f)=p+1$  and
$\rho_{p+1}(f)=\rho _{p}(A_{0})$.
\end{theorem}

\begin{theorem}[\cite{t2}] \label{thmB}
Let $A_{0}(z),\dots$, $A_{k-1}(z)$
be entire functions such that $\rho (A_{0})=\rho $
$(0<\rho <+\infty )$  and $\tau (A_{0})=\tau $
$(0<\tau <+\infty )$,  and let
$\rho (A_{j})\leq \rho (A_{0})=\rho $ $(j=1,2,\dots,k-1)$,
$\tau (A_{j})<\tau (A_{0})=\tau $ $(j=1,2,\dots,k-1)$
 if $\rho (A_{j})=\rho (A_{0})$ $(j=1,2,\dots,k-1)$.
 Then every solution $f\not\equiv 0$  of
\eqref{e1.4} satisfies $\rho (f)=+\infty $
and $\rho _{2}(f)=\rho (A_{0})=\rho $.
\end{theorem}

 The purpose of this paper is to extend and to improve the
results of Theorems \ref{thmA} and \ref{thmB}.
We will prove the following theorems:


\begin{theorem} \label{thm1.1}
Let $A_{0}(z),\dots, A_{k-1}(z)$  be entire functions, and let
$i(A_{0})=p$ $(0<p<\infty )$.
Assume that $\max \{\rho _{p}(A_{j}):j=1,2,\dots,k-1\}\leq \rho
_{p}(A_{0})=\rho $ $(0<\rho <+\infty )$
 and $\max \{\tau _{p}(A_{j}):\rho _{p}( A_{j})=\rho _{p}(A_{0})\}
<\tau _{p}(A_{0}) =\tau $ $(0<\tau <+\infty )$.
 Then every solution $f\not\equiv 0$  of \eqref{e1.4}
satisfies $i(f)=p+1$ and $\rho _{p+1}(f)=\rho _{p}(A_{0})=\rho $.
\end{theorem}

 Combining Theorems \ref{thmA} and \ref{thm1.1}, we obtain the following
result.


\begin{corollary} \label{coro1.1}
Let $A_{0}(z),\dots, A_{k-1}(z)$ be entire functions, and let
$i(A_{0})=p$ $(0<p<\infty )$.
Assume that either
\begin{equation*}
\max \{i(A_{j}):j=1,2,\dots,k-1\}<p
\end{equation*}
or
\begin{equation*}
\max \{\rho _{p}(A_{j}):j=1,2,\dots,k-1\}\leq \rho _{p}(
A_{0})=\rho \quad(0<\rho <+\infty )
\end{equation*}
and
\begin{equation*}
\max \{\tau _{p}(A_{j}):\rho _{p}(A_{j})=\rho_{p}(A_{0})\}
<\tau _{p}(A_{0})=\tau  \quad (0<\tau <+\infty ).
\end{equation*}
 Then every solution $f\not\equiv 0$  of \eqref{e1.4}
 satisfies $i(f)=p+1$  and $\rho_{p+1}(f)=\rho _{p}(A_{0})=\rho $.
\end{corollary}

\begin{corollary} \label{coro1.2}
Let $A_{0}(z),\dots, A_{k-1}(z)$  be entire functions, and let
$i(A_{j})=p$ $(j=0,\dots,k-1)$,
$(0<p<\infty )$.  Assume that $\rho _{p}(A_{j})=\rho $
$(j=0,\dots,k-1)$,
$(0<\rho <+\infty )$  and
$\max \{\tau _{p}( A_{j}): j=1,2,\dots,k-1\}<\tau _{p}(A_{0})
=\tau $ $(0<\tau <+\infty )$.
Then every solution $f\not\equiv 0$  of \eqref{e1.4}
satisfies $i(f)=p+1$  and $\rho _{p+1}(f)=\rho $.
\end{corollary}

\begin{corollary} \label{coro1.3}
Let $h_{j}(z)$  $(j=0,1,\dots,k-1)$ $(k\geq 2)$
 be entire functions with $h_{0}\not\equiv 0$,
$\rho _{p}( h_{j})<n$ $(0<p<\infty )$, and let
$A_{j}(z)=h_{j}(z)\exp _{p}(P_{j}(z))$, where
$P_{j}(z)=\sum_{i=0}^n a_{ji}z^{i}$ $(j=0,\dots,k-1)$
 are polynomials with degree $n\geq 1$, $a_{jn}$
$(j=0,1,\dots,k-1)$ are complex numbers. If
$|a_{0n}| >\max \{ |a_{jn}| :j=1,\dots,k-1\}$,
 then every solution $f\not\equiv 0$
 of \eqref{e1.4} satisfies $i(f)=p+1$
 and $\rho _{p+1}(f)=n$.
\end{corollary}

 Replacing the dominant coefficient $A_{0}$ by an arbitrary
coefficient $A_{s}$, where $s\in \{ 1,\dots,k-1\} $, we obtain the
following result which is an extension of Theorems \ref{thmA},
\ref{thmB} and  \ref{thm1.1}.


\begin{theorem} \label{thm1.2}
Let $A_{0}(z),\dots, A_{k-1}(z)$ be entire functions. Suppose
that there exists an $A_{s}$ $(1\leq s\leq k-1)$
 with $i(A_{s})=p$ $(0<p<\infty)$. Assume that
$\max \{\rho _{p}(A_{j}): j\neq s\}\leq \rho _{p}(A_{s})=\rho $
$(0<\rho <+\infty )$  and
$\max \{\tau _{p}( A_{j}):\rho _{p}(A_{j})=\rho _{p}(
A_{s})\}<\tau _{p}(A_{s})=\tau $
$(0<\tau <+\infty )$.  Then every transcendental solution
$f$ of \eqref{e1.4}  satisfies $p\leq i(f)\leq p+1$
 and $\rho _{p+1}(f)\leq \rho_{p}(A_{s})\leq \rho _{p}(f)$,
and every non-transcendental solution $f$  of \eqref{e1.4}
is a polynomial of degree $\deg(f)\leq s-1$.
\end{theorem}

 Many important results have been obtained on the fixed points of
general transcendental meromorphic functions for almost four decades
(see \cite{z1}). However, there are a few studies on the fixed points of
solutions of differential equations. It was in year 2000 that  Chen
first pointed out the relation between the exponent of convergence of
distinct fixed points and the rate of growth of solutions of second order
linear differential equations with entire coefficients
(see \cite{c4}).

 In the following, we investigate the relationship between
solutions of the differential equation \eqref{e1.4} and entire
functions with finite iterated $p$-order. We obtain some precise estimates
of their fixed points.

To give the precise estimate of fixed points, we define the following
concept.

\begin{definition}[\cite{k1,l2}] \label{def1.4} \rm
 Let $f$ be a meromorphic function.
Then the iterated exponent of convergence of the sequence of distinct
zeros of $f(z)$ is defined by
\begin{equation}
\overline{\lambda }_{p}(f)=\limsup_{r\to +\infty}
\frac{\log _{p}\overline{N}(r,\frac{1}{f})}{\log r}\quad
(1\leq p\text{ an integer}),  \label{e1.5}
\end{equation}
where $\overline{N}(r,1/f)$ is the counting function of
distinct zeros of $f(z)$ in $\{ |z|<r\}$. For $p=1$,
this notation is called exponent of convergence of
the sequence of distinct zeros and for $p=2$ hyper-exponent of convergence
of the sequence of distinct zeros.
\end{definition}

\begin{definition}[\cite{l2}] \label{def1.5} \rm
Let $f$ be a meromorphic function. Then the iterated
exponent of convergence of the sequence of distinct fixed points of
$f(z)$ is defined by
\begin{equation}
\overline{\lambda }_{p}(f-z)=\limsup_{r\to +\infty}
\frac{\log _{p}\overline{N}(r,\frac{1}{f-z})}{
\log r}\quad (1\leq p\text{ an integer}).  \label{e1.6}
\end{equation}
For $p=1$, this notation is called exponent of convergence of the
sequence of distinct fixed points and for $p=2$ hyper-exponent
of convergence of the sequence of distinct fixed points
(see \cite{l3,w1}). Thus $\overline{\lambda }_{p}(f-z)$
is an indication of oscillation of distinct fixed points of
$f(z)$.
\end{definition}


 We obtain the following results.


\begin{theorem} \label{thm1.3}
Suppose that $A_{0}(z),\dots,A_{k-1}(z)$ satisfy the hypotheses of
Theorem \ref{thm1.1}, and let $\varphi \not\equiv 0$ be a finite iterated
$p$-order entire function. Then every solution $f\not\equiv 0$
of \eqref{e1.4} satisfies
\[
\overline{\lambda }_{p}(f-\varphi )=\lambda _{p}(
f-\varphi )=\rho _{p}(f)=+\infty
\]
and $\overline{\lambda }_{p+1}(f-\varphi )=\lambda _{p+1}(
f-\varphi )=\rho _{p+1}(f)=\rho $.
\end{theorem}

 Setting $\varphi (z)=z$ in Theorem \ref{thm1.3}, we obtain the
following corollary.


\begin{corollary} \label{coro1.4}
Under the hypotheses of Theorem \ref{thm1.3},
every solution $f\not\equiv 0$  of the equation
\eqref{e1.4} satisfies
$\overline{\lambda }_{p}(f-z)=\lambda _{p}(f-z)=\rho _{p}(f)=+\infty$
and $\overline{\lambda }_{p+1}(f-z)=\lambda_{p+1}(f-z)=\rho _{p+1}(f)
=\rho $.
\end{corollary}

\begin{corollary} \label{coro1.5}
Suppose that $A_{j}(z)=h_{j}(z)\exp _{p}(P_{j}(z))$
$(j=0,1,\dots,k-1)$ and the complex numbers $a_{jn}$
$(j=0,1,\dots,k-1)$  satisfy the hypotheses of Corollary \ref{coro1.3}.
If $\varphi \not\equiv 0$  is a finite iterated
$p$-order entire function, then every solution
$f\not\equiv 0$  of \eqref{e1.4} satisfies
$\overline{\lambda }_{p}(f-\varphi )=\lambda _{p}(
f-\varphi )=\rho _{p}(f)=+\infty $ and
$\overline{\lambda }_{p+1}(f-\varphi )=\lambda _{p+1}(
f-\varphi )=\rho _{p+1}(f)=n$. In particular,
every solution $f(z)\not\equiv 0$ of
\eqref{e1.4} satisfies
$\overline{\lambda }_{p}(f-z)=\lambda _{p}(f-z)=\rho _{p}(
f)=+\infty $  and $\overline{\lambda }_{p+1}(f-z)
=\lambda _{p+1}(f-z)=\rho _{p+1}(f)=n$.
\end{corollary}

\section{Preliminary Lemmas}

 Our proofs depend mainly upon the following lemmas. Before
starting these lemmas, we recall the concepts of linear and logarithmic
measure. For $E\subset [0,+\infty )$, we define the linear
measure of a set $E$ by
$ m (E)=\int_{0}^{+\infty }\chi _{E}(t)dt$ and the
logarithmic measure of a set $F\subset [1,+\infty )$ by
$\mathop{\rm lm}(F)=\int_{1}^{+\infty }(\chi _{F}(t)/t)dt$, where
$\chi _{H}$ is the characteristic function of a set $H$.

\begin{lemma}[{\cite[p. 90]{g1}}] \label{lem2.1}
Let $f(z)$ be a transcendental meromorphic function, and let
$\alpha >1$ be a given constant. Then there exist a set
$E_{1}\subset $ $(1,+\infty )$ of finite logarithmic measure and
a constant $B>0$ that
depends only on $\alpha $ and $(m,n)$
($m,n$ positive integers with $m<n$) such that for all $z$
satisfying $|z| =r\notin [0,1] \cup E_{1}$, we have
\begin{equation}
\Big|\frac{f^{(n)}(z)}{f^{(m)
}(z)}\Big| \leq B\Big[\frac{T(\alpha r,\,f)
}{r}(\log ^{\alpha }r)\log T(\alpha r,\,f)\Big]
^{n-m}.  \label{e2.1}
\end{equation}
\end{lemma}

 \begin{lemma} \label{lem2.2}
 Let $f(z)$ be an entire function of iterated $p$-order
$0<\rho _{p}(f)<+\infty $ and iterated $p$-type
$0<\tau _{p}(f)<\infty $. Then for any given
$\beta <\tau_{p}(f)$,  there exists a set
$E_{2}\subset [ 1,+\infty )$ that has infinite logarithmic measure, such
that for all $r\in E_{2}$,  we have
\begin{equation}
\log _{p}M(r,f)>\beta r^{\rho _{p}(f)}.  \label{e2.2}
\end{equation}
\end{lemma}

 \begin{proof} When $p=1$, the Lemma is due to  Tu and  Yi \cite{t2}.
 Thus we assume that $p\geq 2$. By definitions of
iterated order and iterated type, there exists an increasing sequence
$\{ r_{n}\} $, $r_{n}\to \infty $ satisfying
$(1+ \frac{1}{n})r_{n}<r_{n+1}$ and
\begin{equation}
\lim_{r_{n}\to +\infty } \frac{\log _{p}M(
r_{n},f)}{r_{n}^{\rho _{p}(f)}}=\tau _{p}(f)
>\beta .  \label{e2.3}
\end{equation}
Then there exists a positive integer $n_{0}$ such that for all
$n>n_{0}$ and
for any given $\varepsilon $ $(0<\varepsilon <\tau _{p}(f)
-\beta )$, we have
\begin{equation}
\log _{p}M(r_{n},f)>(\tau _{p}(f)
-\varepsilon )r_{n}^{\rho _{p}(f)}.  \label{e2.4}
\end{equation}
For any given $\beta <\tau _{p}(f)$, there exists a positive
integer $n_{1}$ such that for all $n>n_{1}$, we have
\begin{equation}
\big(\frac{n}{n+1}\big)^{\rho _{p}(f)}>\frac{\beta }{\tau
_{p}(f)-\varepsilon }\,.  \label{e2.5}
\end{equation}
Taking $n\geq n_{2}=\max\{ n_{0},n_{1}\} $. By
\eqref{e2.4} and \eqref{e2.5} for any $r\in [r_{n},(1+
\frac{1}{n})r_{n}] $, we obtain
\begin{equation}
\begin{aligned}
\log _{p}M(r,f)
&\geq \log _{p}M(r_{n},f)\\
&>(\tau _{p}(f)-\varepsilon )r_{n}^{\rho _{p}(f)}\\
&\geq (\tau _{p}(f)-\varepsilon )\big(\frac{n}{n+1
}r\big)^{\rho _{p}(f)}\\
&>\beta r^{\rho _{p}(f)}.
\end{aligned}\label{e2.6}
\end{equation}
Set $E_{2}=\cup_{n=n_{2}}^{\infty } [r_{n},(1+
\frac{1}{n})r_{n}] $, then there holds
\begin{equation}
\mathop{\rm lm}(E_{2})=\sum_{n=n_{2}}^\infty
\int_{r_{n}}^{(1+\frac{1}{n})r_{n}} \frac{dt}{t}
= \sum_{n=n_{2}}^{\infty } \log (1+\frac{1}{n})
=+\infty .  \label{e2.7}
\end{equation}
\end{proof}

Using similar arguments as in the proof of Lemma \ref{lem2.2}, we
obtain the following result.

 \begin{lemma} \label{lem2.3}
Let $f(z)$ be an entire function of iterated $p$-order
$0<\rho _{p}(f)<+\infty $ and iterated $p$-type
$0<\tau_{p}(f)<\infty $.  Then for any given
$\beta <\tau _{p}(f)$,  there exists a set
$E_{3}\subset [1,+\infty )$  that has infinite logarithmic measure, such
that for all $r\in E_{3}$,  we have
\begin{equation}
\log _{p-1} m (r,f)=\log _{p-1}T(r,f)>\beta r^{\rho
_{p}(f)}.  \label{e2.8}
\end{equation}
\end{lemma}

To avoid some problems caused by the exceptional set we
recall the following Lemmas.


 \begin{lemma}[{\cite[p. 68]{b1}}] \label{lem2.4}
Let $g:[0,+\infty )\to \mathbb{R}$  and
$h:[0,+\infty )\to \mathbb{R}$ be monotone non-decreasing functions
such that $g(r)\leq h(r)$ outside of an exceptional set $E_{4}$
of finite linear measure. Then for any $\alpha >1$, there
exists $r_{0}>0$ such that $g(r)\leq h(\alpha r)$ for all $r>r_{0}$.
\end{lemma}

 \begin{lemma}[\cite{g2}] \label{lem2.5}
Let $\varphi :[0,+\infty )\to \mathbb{R}$ and
$\psi :[0,+\infty )\to \mathbb{R}$ be monotone non-decreasing functions
such that $\varphi (r)\leq \psi (r)$ for all
$r\notin E_{5}\cup [0,1] $, where $E_{5}\subset (1,+\infty )$
 is a set of finite logarithmic measure. Let $\gamma >1$
be a given constant. Then there exists an
$r_{1}=r_{1}(\gamma )>0$ such that
$\varphi (r)\leq \psi (\gamma r)$  for all $r>r_{1}$.
\end{lemma}

\begin{lemma}[\cite{b5,c1}] \label{lem2.6}
Let $A_{0}$, $A_{1},\dots, A_{k-1}$,
$F\not\equiv 0$ be finite iterated $p$-order meromorphic
functions. If $f$ is a meromorphic solution with
$\rho _{p}(f)=+\infty $ and $\rho _{p+1}(f)=\rho <+\infty $
of the equation
\begin{equation}
 f^{(k)}+A_{k-1}f^{(k-1)}+\dots+A_{1}f'+A_{0}f=F,  \label{e2.9}
\end{equation}
then $\overline{\lambda }_{p}(f)=\lambda _{p}(
f)=\rho _{p}(f)=+\infty $ and
$\overline{ \lambda }_{p+1}(f)=\lambda _{p+1}(f)=\rho
_{p+1}(f)=\rho $.
\end{lemma}

\begin{lemma}[\cite{b6,l1}] \label{lem2.7}
Suppose that $k\geq 2$ and $A_{0}( z),\dots,A_{k-1}(z)$
are entire functions of finite iterated $p$-order. If $f(z)$
is a solution of \eqref{e1.4}, then
$i(f)\leq p+1$ and $\rho _{p+1}(f)\leq \max\{
\rho _{p}(A_{j}): j=0,\dots,k-1\} =\rho $.
\end{lemma}

\section{Proof of Theorem \ref{thm1.1}}

 Suppose that $f\not\equiv 0$ is a solution of
equation \eqref{e1.4}. From \eqref{e1.4}, we can write
\begin{equation}
|A_{0}(z)| \leq \Big|\frac{f^{(
k)}}{f}\Big| +|A_{k-1}(z)|
\Big|\frac{f^{(k-1)}}{f}\Big| +\dots+|
A_{1}(z)| \big|\frac{f'}{f}\big|.  \label{e3.1}
\end{equation}
By Lemma \ref{lem2.1}, there exist a constant $B>0$ and a set
$E_{1}\subset (1,+\infty )$ having finite logarithmic measure
such that for all $z$
satisfying $|z| =r\notin E_{1}\cup [0,1] $,
we have
\begin{equation}
\Big|\frac{f^{(j)}(z)}{f(z)}\Big| \leq B[
T(2r,f)] ^{k+1} \quad (j=1,2,\dots,k).  \label{e3.2}
\end{equation}
If $\max \{\rho _{p}(A_{j}):j=1,2,\dots,k-1\}<\rho _{p}(
A_{0})=\rho $, then by Theorem \ref{thmA}, we obtain $i(f)=p+1$
and $\rho _{p+1}(f)=\rho _{p}(A_{0})=\rho $.

 If $\max \{\rho _{p}(A_{j}):j=1,2,\dots,k-1\}=\rho
_{p}(A_{0})=\rho$  $(0<\rho <+\infty )$ and
 $\max \{\tau _{p}(A_{j}):\rho _{p}(A_{j})=\rho _{p}(A_{0})\}
<\tau _{p}(A_{0}) =\tau $ $(0<\tau <+\infty )$. Then, there exists a set
$I\subseteq\{1,2,\dots,k-1\}$ such that
$\rho _{p}( A_{j})=\rho _{p}(A_{0})=\rho (j\in I)$ and
$\tau _{p}(A_{j})<\tau _{p}(A_{0})$
$(j\in I)$. Thus, we choose $\alpha _{1}$, $\alpha _{2}$
satisfying $\max \{\tau _{p}(A_{j}) : (j\in I)
\}<\alpha _{1}<\alpha _{2}<\tau _{p}(A_{0})=\tau $ such that
for sufficiently large $r$, we have
\begin{gather}
|A_{j}(z)| \leq \exp _{p}(\alpha
_{1}r^{\rho }) \quad (j\in I), \label{e3.3}
\\
|A_{j}(z)| \leq \exp _{p}(r^{\alpha
_{0}}) \quad (j\in\{ 1,2,\dots,k-1\} \backslash
I),  \label{e3.4}
\end{gather}
where $0<\alpha _{0}<\rho $. By Lemma \ref{lem2.2}, there exists a set
$E_{2}\subset [1,+\infty )$ with infinite logarithmic measure
such that for all $r\in E_{2}$, we have
\begin{equation}
M(r,A_{0})>\exp _{p}(\alpha _{2}r^{\rho }).
\label{e3.5}
\end{equation}
Hence from \eqref{e3.1}-\eqref{e3.5}, for all $z$ satisfying
$|A_{0}(z)| =M(r,A_{0})$ and
for sufficiently large $|z| =r\in E_{2}\backslash E_{1}\cup [0,1] $,
we have
\begin{equation}
\exp _{p}(\alpha _{2}r^{\rho })\leq kB\exp _{p}(\alpha
_{1}r^{\rho })[T(2r,f)] ^{k+1}.  \label{e3.6}
\end{equation}
Since $\alpha _{2}>\alpha _{1}>0$, we get from \eqref{e3.6} that
\begin{equation}
\exp ((1-\gamma )\exp _{p-1}(\alpha _{2}r^{\rho
}))\leq kB[T(2r,f)] ^{k+1}\text{,}  \label{e3.7}
\end{equation}
where $\gamma $ $(0<\gamma <1)$ is a real number.
By \eqref{e3.7}, Lemma \ref{lem2.5} and the definition of
iterated order, we have
$i(f)\geq p+1$ and $\rho _{p+1}(f)\geq \rho
_{p}(A_{0})=\rho $.
On the other hand by Lemma \ref{lem2.7}, we have
$i(f)\leq p+1$ and $\rho _{p+1}(f)\leq \rho
_{p}(A_{0})=\rho $, hence $i(f)=p+1$ and $\rho
_{p+1}(f)=\rho _{p}(A_{0})=\rho $.

\section{Proof of Theorem \ref{thm1.2}}

 Assume that $f$ is a transcendental solution of \eqref{e1.4}.
It follows from \eqref{e1.4} that
\begin{equation}
\begin{aligned}
A_{s}(z)&=-\Big(\frac{f^{(k)}}{f^{(s)
}}+A_{k-1}(z)\frac{f^{(k-1)}}{f^{(s)}}
+\dots+A_{s+1}(z)\frac{f^{(s+1)}}{f^{(s)}}
\\
&\quad  +A_{s-1}(z)\frac{f^{(s-1)}}{f^{(
s)}}+\dots+A_{1}(z)\frac{f'}{f^{(s)}}
+A_{0}(z)\frac{f}{f^{(s)}}\Big)
\\
&=-\frac{f}{f^{(s)}}\Big(\frac{f^{(k)}}{f}
+A_{k-1}(z)\frac{f^{(k-1)}}{f}+\dots+A_{s+1}(
z)\frac{f^{(s+1)}}{f}
\\
&\quad +A_{s-1}(z)\frac{f^{(s-1)}}{f}
+\dots+A_{1}(z)\frac{f'}{f}+A_{0}(z)\Big).
\end{aligned}  \label{e4.1}
\end{equation}
By Lemma of logarithmic derivative \cite{h1} and
\eqref{e4.1}, we obtain
\begin{equation}
\begin{aligned}
T(r,A_{s})
&=m(r,A_{s})\\
&\leq m\big(r,\frac{f}{f^{(s)}}\big)
+\sum_{j\neq s} m(r,A_{j})  +O(\log (rT(r,f)))\\
&=m\big(r,\frac{f}{f^{(s)}}\big)+\sum_{j\neq s}
T(r,A_{j})+O(\log (rT(r,f)))
\end{aligned}  \label{e4.2}
\end{equation}
holds for all $r$ outside a set $E\subset (0,+\infty )$ with a
finite linear measure $m(E)<+\infty $. Noting that
\begin{equation}
\begin{aligned}
m(r,\frac{f}{f^{(s)}})
&\leq m(r,f) +m\big(r,\frac{1}{f^{(s)}}\big)\\
&\leq T(r,f)+T(r,\frac{1}{f^{(s)}})\\
&=T(r,f)+T(r,f^{(s)})+O(1)\\
&\leq (s+2)T(r,f)+o(T(r,f) )+O(1).
\end{aligned}  \label{e4.3}
\end{equation}
For sufficiently large $r$, we have
\begin{equation}
O(\log r+\log T(r,f))\leq \frac{1}{2}T(r,f).  \label{e4.4}
\end{equation}
Thus, by \eqref{e4.2}-\eqref{e4.4}, for sufficiently large
$r\notin E$, we have
\begin{equation}
T(r,A_{s})\leq cT(r,f)+\sum_{j\neq s}
T(r,A_{j}),  \label{e4.5}
\end{equation}
where $c$ is a positive constant.

 If $\max \{\rho _{p}(A_{j}):j=0,1,\dots,s-1,s+1,\dots,k-1\}
<\rho _{p}(A_{s})=\rho $, then for
sufficiently large $r$, we have
\begin{equation}
T(r,A_{j})\leq \exp _{p-1}(r^{\beta _{0}}) \quad
(j=0,1,\dots,s-1,s+1,\dots,k-1),  \label{e4.6}
\end{equation}
where $0<\beta _{0}<\rho $. Since $\rho _{p}(A_{s})=\rho $,
there exists $\{ r_{n}'\}$ $(r_{n}'\to +\infty )$ such that
\begin{equation}
\lim_{r_{n}'\to +\infty } \frac{\log _{p}T(r_{n}',A_{s})}{\log r_{n}'}
=\rho .
\label{e4.7}
\end{equation}
Set the linear measure of $E$, $m(E)=\delta <+\infty $, then
there exists a point $r_{n}\in [r_{n}',r_{n}'+\delta +1] -E$. From
\begin{equation}
\frac{\log _{p}T(r_{n},A_{s})}{\log r_{n}}
\geq \frac{\log _{p}T(r_{n}',A_{s})}{\log (r_{n}'+\delta +1)}
=\frac{\log _{p}T(r_{n}',A_{s})}{\log r_{n}'+\log (1+(\delta +1)
/r_{n}')}  \label{e4.8}
\end{equation}
we get
\begin{equation}
\lim_{r_{n}\to +\infty }\frac{\log _{p}T(
r_{n},A_{s})}{\log r_{n}}=\rho .  \label{e4.9}
\end{equation}
So for any given $\varepsilon $
$(0<\varepsilon <\rho -\beta_{0})$, and for $j\neq s$,
\begin{gather}
T(r_{n},A_{j})\leq \exp _{p-1}(r_{n}^{\beta _{0}}) \label{e4.10}
\\
T(r_{n},A_{s})\geq \exp _{p-1}(r_{n}^{\rho -\varepsilon})  \label{e4.11}
\end{gather}
hold for sufficiently large $r_{n}$. By \eqref{e4.5},
\eqref{e4.10}, \eqref{e4.11} and Lemma \ref{lem2.4}, we obtain for
sufficiently large $r_{n}$
\begin{equation}
\exp _{p-1}(r_{n}^{\rho -\varepsilon })\leq cT(
r_{n},f)+(k-1)\exp _{p-1}(r_{n}^{\beta
_{0}}).  \label{e4.12}
\end{equation}
Therefore, by \eqref{e4.12} we obtain $i(f)\geq p$ and
\begin{equation}
\limsup_{r_{n}\to +\infty }
\frac{\log _{p}T( r_{n},f)}{\log r_{n}}\geq \rho -\varepsilon   \label{e4.13}
\end{equation}
and since $\varepsilon >0$ is arbitrary, we get
$\rho _{p}(f) \geq \rho _{p}(A_{s})=\rho $. On the other hand
by Lemma \ref{lem2.7},
we have $i(f)\leq p+1$ and
$\rho _{p+1}(f)\leq \rho _{p}(A_{s})$. Hence, we obtain
$p\leq i(f) \leq p+1$ and $\rho _{p+1}(f)\leq \rho _{p}(A_{s})
\leq \rho _{p}(f)$.

 If $\max \{\rho _{p}(A_{j}):j=0,1,\dots,s-1,s+1,\dots,k-1\}
=\rho _{p}(A_{s})=\rho$ $(0<\rho <+\infty )$ and
$\max \{\tau _{p}(A_{j}):\rho _{p}(A_{j})=\rho _{p}(A_{s})
\}<\tau _{p}(A_{s})=\tau $ $(0<\tau <+\infty )$.
Then, there exists a set $J\subseteq\{ 0,1,\dots,s-1,s+1,\dots,k-1\}$
 such that $\rho _{p}(A_{j})=\rho _{p}(A_{s})=\rho$
$(j\in J)$ and $\tau _{p}(A_{j})<\tau _{p}(A_{s})$ $(j\in J)$.
Thus, we choose $\beta _{2}$, $\beta _{3}$ satisfying
$\max \{\tau _{p}(A_{j}):(j\in J)\}<\beta _{2}<\beta _{3}<\tau _{p}(A_{s})
=\tau $ such that for sufficiently large $r$, we have
\begin{gather}
T(r,A_{j})\leq \exp _{p-1}(\beta _{2}r^{\rho }) \quad (j\in J),
  \label{e4.14}\\
T(r,A_{j})\leq \exp _{p-1}(r^{\beta _{1}})\quad
(j\in\{ 0,1,\dots,s-1,s+1,\dots,k-1\} \backslash J),
\label{e4.15}
\end{gather}
where $0<\beta _{1}<\rho $. By Lemma \ref{lem2.3}, there exists a set
$E_{3}\subset [1,+\infty )$ with infinite logarithmic measure
such that for all $r\in E_{3}$, we have
\begin{equation}
T(r,A_{s})>\exp _{p-1}(\beta _{3}r^{\rho }).
\label{e4.16}
\end{equation}
Hence from \eqref{e4.5}, \eqref{e4.14}, \eqref{e4.15},
 \eqref{e4.16} and Lemma \ref{lem2.4}, for sufficiently large
$|z| =r\in E_{3}$, we have
\begin{equation}
\exp _{p-1}(\beta _{3}r^{\rho })\leq cT(r,f)
+(k-1)\exp _{p-1}(\beta _{2}r^{\rho }).  \label{e4.17}
\end{equation}
By this inequality  and the definition of iterated order,
we have $i(f)\geq p$ and $\rho _{p}(f)\geq \rho _{p}(
A_{s})=\rho $. On the other hand by Lemma \ref{lem2.7},
we have $i(f)\leq p+1$ and $\rho _{p+1}(f)\leq \rho _{p}(
A_{s})$. Hence, we obtain $p\leq i(f)\leq p+1$ and
$\rho_{p+1}(f)\leq \rho _{p}(A_{s})\leq \rho _{p}(f)$.

 Suppose that $f$ is a polynomial of $\deg f=m\geq s$. If
$\max \{\rho _{p}(A_{j}):j=0,1,\dots,s-1,s+1,\dots,k-1\}<\rho
_{p}(A_{s})=\rho $, then
\begin{equation}
i(0)=i(f^{(k)}+A_{k-1}(z)
f^{(k-1)}+\dots+A_{1}(z)f'+A_{0}(
z)f)=i(A_{s}) \label{e4.18}
\end{equation}
and
\begin{equation}
\rho _{p}(0)=\rho _{p}(f^{(k)}+A_{k-1}(
z)f^{(k-1)}+\dots+A_{1}(z)f'+A_{0}(z)f)=\rho _{p}(A_{s})>0  \label{e4.19}
\end{equation}
this is a contradiction by \eqref{e1.4}.

 If $\max \{\rho _{p}(A_{j}):j=0,1,\dots,s-1,s+1,\dots,k-1\}
=\rho _{p}(A_{s})=\rho$
$(0<\rho <+\infty )$ and
$\max \{\tau _{p}( A_{j}):\rho _{p}(A_{j})=\rho _{p}(A_{s})
\}<\tau _{p}(A_{s})=\tau $ $(0<\tau <+\infty )$.
Then, there exists a set
$K\subseteq\{ 0,1,\dots,s-1,s+1,\dots,k-1\}$ such that
$\rho _{p}(A_{j})=\rho _{p}(A_{s})=\rho \mathit{\ }(j\in K)$
and $\tau _{p}(A_{j}) <\tau _{p}(A_{s})$ $(j\in K)$. Thus, we choose
$\beta _{4}$, $\beta _{5}$ satisfying
$\max \{\tau _{p}(A_{j}): (j\in K)\}<\beta _{4}<\beta _{5}<\tau _{p}(
A_{s})=\tau $ such that for sufficiently large $r$, we have
\begin{gather}
|A_{j}(z)| \leq \exp _{p}(\beta_{4}r^{\rho })
\quad (j\in K), \label{e4.20} \\
|A_{j}(z)| \leq \exp _{p}(r^{\beta
_{6}}) \quad (j\in\{ 0,1,\dots,s-1,s+1,\dots,k-1\}
\backslash K), \label{e4.21}
\end{gather}
where $0<\beta _{6}<\rho $. By Lemma \ref{lem2.2}, there exists a set
$E_{2}\subset [1,+\infty )$ with infinite logarithmic measure
such that for all $r\in E_{2}$, we have
\begin{equation}
M(r,A_{s})>\exp _{p}(\beta _{5}r^{\rho }).
\label{e4.22}
\end{equation}
Hence from \eqref{e1.4}, \eqref{e4.20}-\eqref{e4.22},
for all $z$ satisfying $|A_{s}(z)| =M(r,A_{s})$ and for sufficiently
large $|z| =r\in E_{2}$, we have
\begin{equation}
\begin{aligned}
d_{1}r^{m-s}\exp _{p}(\beta _{5}r^{\rho })
&\leq | A_{s}(re^{i\theta })f^{(s)}(re^{i\theta })|\\
&\leq \sum_{j\neq s}|A_{j}(re^{i\theta })f^{(j)}(re^{i\theta })| \\
&\leq d_{2}r^{m}\exp _{p}(\beta _{4}r^{\rho }),
\end{aligned}  \label{e4.23}
\end{equation}
where $d_{1},d_{2}$ are positive constants. By
\eqref{e4.23}, we get
\begin{equation}
\exp\{ \exp _{p-1}(\beta _{5}r^{\rho })-\exp _{p-1}(
\beta _{4}r^{\rho })\} \leq \frac{d_{2}}{d_{1}}r^{s}.
\label{e4.24}
\end{equation}
Hence by $\beta _{5}>\beta _{4}>0$, from \eqref{e4.24} we obtain a
contradiction. Therefore, if $f$ is a non-transcendental solution, then it
must be a polynomial of $\deg f\leq s-1$. This proves
Theorem \ref{thm1.2}.

\section{Proof of Theorem \ref{thm1.3}}

 Suppose that $f\not\equiv 0$ is a solution of
equation \eqref{e1.4}. Then by Theorem \ref{thm1.1}, we have
$\rho _{p}( f)=\infty $ and
$\rho _{p+1}(f)=\rho _{p}( A_{0})=\rho $. Set
$w=f-\varphi $. Since $\rho _{p}(\varphi)<\infty $, then we
have $\rho _{p}(w)=\rho _{p}(f-\varphi )=\rho _{p}(f)=\infty $
and $\rho _{p+1}(w)=\rho _{p+1}(f-\varphi )=\rho _{p+1}(f)
=\rho _{p}(A_{0})=\rho $. Substituting $f=w+\varphi $ into
equation \eqref{e1.4}, we obtain
\begin{equation}
\begin{aligned}
w^{(k)}+A_{k-1}(z)w^{(k-1)}+\dots+A_{0}(z)w
&=-(\varphi^{(k)}+A_{k-1}(z)\varphi
^{(k-1)}+\dots+A_{0}(z)\varphi)\\ &=W.
\end{aligned} \label{e5.1}
\end{equation}
Since $\varphi \not\equiv 0$ and $\rho _{p}(\varphi )<\infty$,
 we have $W\not\equiv 0$. Then by Lemma \ref{lem2.6}, we obtain
$\overline{\lambda }_{p}(w)=\lambda _{p}(w)
=\rho _{p}(w)=\infty $ and
$\overline{\lambda }_{p+1}(w)=\lambda_{p+1}(w)
=\rho _{p+1}(w)=\rho _{p}(A_{0})=\rho $; i.e.,
$\overline{\lambda }_{p}(f-\varphi)=\lambda _{p}(f-\varphi )
=\rho _{p}(f) =\infty $ and $\overline{\lambda }_{p+1}(f-\varphi )
=\lambda_{p+1}(f-\varphi )=\rho _{p+1}(f)=\rho _{p}(A_{0})=\rho $.

\subsection*{Acknowledgements}
The author would like to thank the anonymous
referees for their helpful remarks and suggestions to improve
this article.

\begin{thebibliography}{00}

\bibitem{b1} S. Bank;
 \textit{General theorem concerning the
growth of solutions of first-order algebraic differential equations},
Compositio Math. 25 (1972), 61--70.

\bibitem{b2} B. Bela\"{\i}di;
\textit{On the iterated order and
the fixed points of entire solutions of some complex linear differential
equations}, Electron. J. Qual. Theory Differ. Equ. 2006, No. 9, 11 pp.

\bibitem{b3} B. Bela\"{\i}di;
\textit{Iterated order of fast
growth solutions of linear differential equations}, Aust. J. Math. Anal.
Appl. 4 (2007), no. 1, Art. 20, 8 pp.

\bibitem{b4} B. Bela\"{\i}di; \textit{Growth and
oscillation theory of solutions of some linear differential equations}, Mat.
Vesnik 60 (2008), no. 4, 233--246.

\bibitem{b5} B. Bela\"{\i}di; \textit{Oscillation of fixed
points of solutions of some linear differential equations}, Acta Math. Univ.
Comenian. (N.S.) 77 (2008), no. 2, 263--269.

\bibitem{b6} L. G. Bernal;
 \textit{On growth }$k$\textit{-order
of solutions of a complex homogeneous linear differential equation}, Proc.
Amer. Math. Soc. 101 (1987), no. 2, 317--322.

\bibitem{c1}T. B. Cao, Z. X. Chen, X. M. Zheng and J. Tu;
\textit{On the iterated order of meromorphic solutions of higher order
linear differential equations}, Ann. Differential Equations
21 (2005), no. 2, 111--122.

\bibitem{c2} T. B. Cao, J. F. Xu and Z. X. Chen;
\textit{On the meromorphic solutions of linear differential equations
on the complex plane}, J. Math. Anal. Appl. To appear.

\bibitem{c3} T. B. Cao and H. X. Yi;
 \textit{On the complex
oscillation of higher order linear differential equations with meromorphic
coefficients}, J. Syst. Sci. Complex. 20 (2007), no. 1, 135--148.

\bibitem{c4} Z. X. Chen; \textit{The fixed points and
hyper order of solutions of second order complex differential equations},
Acta Math. Sci. Ser. A Chin. Ed. 20 (2000), no. 3, 425--432 (in Chinese).

\bibitem{c5} Y. M. Chiang and W. K. Hayman;
 \textit{ Estimates on the growth of meromorphic solutions of linear differential
equations}, Comment. Math. Helv. 79 (2004), no. 3, 451--470.

\bibitem{f1} M. Frei;
 \textit{Sur l'ordre des solutions enti
\`{e}res d'une \'{e}quation diff\'{e}rentielle lin\'{e}aire}, C.
R. Acad. Sci. Paris 236, (1953), 38--40.

\bibitem{g1} G. Gundersen; \textit{Estimates for the
logarithmic derivative of a meromorphic function, plus similar estimates},
J. London Math. Soc. (2) 37 (1988), no. 1, 88--104.

\bibitem{g2} G. G. Gundersen; \textit{Finite order solutions of
second order linear differential equations}, Trans. Amer. Math. Soc. 305
(1988), no. 1, 415--429.

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

\bibitem{k1} L. Kinnunen;
 \textit{Linear differential equations
with solutions of finite iterated order}, Southeast Asian Bull. Math. 22
(1998), no. 4, 385--405.

\bibitem{l1} I. Laine;
 \textit{Nevanlinna theory and complex differential equations},
de Gruyter Studies in Mathematics, 15.
Walter de Gruyter \& Co., Berlin, 1993.

\bibitem{l2} I. Laine and J. Rieppo;
\textit{ Differential polynomials generated by linear differential
equations}, Complex Var. Theory Appl. 49 (2004), no. 12, 897--911.

\bibitem{l3} M. S. Liu and X. M. Zhang;
\textit{Fixed points of meromorphic solutions of higher order
linear differential equations}, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 191--211.

\bibitem{n1} R.  Nevanlinna;
\textit{Eindeutige analytische Funktionen}, Zweite Auflage. Reprint.
Die Grundlehren der mathematischen Wissenschaften,
Band 46. Springer-Verlag, Berlin-New York, 1974.

\bibitem{t1} J. Tu, Z. X. Chen and X. M. Zheng;
\textit{Growth of solutions of complex differential equations with
coefficients of finite iterated order}, Electron. J. Differential
Equations 2006, No. 54, 8 pp.

\bibitem{t2} J. Tu and C. F. Yi; \textit{On the growth of
solutions of a class of higher order linear differential equations with
coefficients having the same order}, J. Math. Anal. Appl. 340 (2008), no. 1,
487--497.

\bibitem{w1} J. Wang and H. X. Yi; \textit{Fixed points and
hyper order of differential polynomials generated by solutions of
differential equation}, Complex Var. Theory Appl. 48 (2003), no. 1, 83--94.

\bibitem{y1} H. X. Yi and C. C. Yang;
\textit{ Uniqueness theory of meromorphic functions}, Mathematics and its
Applications, 557. Kluwer Academic Publishers Group, Dordrecht, 2003.

\bibitem{z1} Q. T. Zhang and C. C. Yang;
\textit{The Fixed Points and Resolution Theory of Meromorphic Functions},
Beijing University Press, Beijing, 1988 (in Chinese).

\end{thebibliography}
\end{document}