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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 81, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/81\hfil Complex oscillation of entire solutions]
{Complex oscillation of entire solutions of
higher-order linear differential equations}

\author[T.-B. Cao\hfil EJDE-2006/81\hfilneg]
{Ting-Bin Cao}

\address{Ting-Bin Cao \newline
Department of Mathematics,  Shandong University,
Jinan, 250100, China}
\email{ctb97@163.com}

\date{}
\thanks{Submitted February 21, 2006. Published July 19, 2006.}
\thanks{Supported by grants 10371065 from the NNSF of China,
 and Z2002A01 from the \hfill\break\indent
NSF of Shandong Province}
\subjclass[2000]{34M10, 30D35}
\keywords{Linear differential equation; meromorphic function;
 iterated order; \hfill\break\indent iterated convergence exponent}


\begin{abstract}
 In this paper, we investigate higher-order linear differential
 equations with entire coefficients of iterated order.
 We improve and extend a result of Bela\"{\i}di and  Hamouda
 by using the estimates for the logarithmic derivative
 of a transcendental meromorphic function due to Gundersen and the
 Winman-Valiron theory. We also consider the nonhomogeneous linear
 differential equations by using  the basic method  and some
 lemmas from the present and other three authors.
\end{abstract}

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

\section{Introduction and main results}

In this paper, we shall assume that the reader is  familiar with
the fundamental results and  the standard notation of the
Nevanlinna value distribution theory of meromorphic functions (see
\cite{yang},\cite{hayman}). The term ``meromorphic function'' will
mean meromorphic in the whole complex plane
$\mathbb{C}$.

For $k\geq 2$, we consider a linear differential equation
\begin{equation}
f^{(k)}+A_{k-1}f^{(k-1)}+\dots+A_{0}f=0, \label{e1.1}
\end{equation}
where $A_{0},\dots,A_{k-1}$ are entire functions with
$A_{0}\not\equiv 0$. It is well known that all solutions of
\eqref{e1.1} are entire functions, and if some of the coefficients
of \eqref{e1.1} are transcendental, then \eqref{e1.1} has at least
one solution with order $\sigma(f)=\infty$.

Thus the question which arises is: What conditions on
$A_{0},\dots,A_{k-1}$ will guarantee that every solution
$f\not\equiv 0$ of \eqref{e1.1} has infinite order?

For the above question, there are many results for second order
linear differential equations (see for example
\cite{frei,gundersen-2,chen-yang-2,chen}). In
2002,  Bela\"{\i}di and  Hamouda considered the higher order
linear differential equations and obtained the following
result.

\begin{theorem}[Bela\"{\i}di and Hamouda \cite{belaidi-hamouda}]
\label{thm1}
 Suppose that there exist
a positive number $\mu$, and a sequence of points $(z_{j})_{j\in
\mathbb{N}}$ with $\lim_{j\to\infty}z_{j}=\infty$,
and two real numbers $\alpha, \beta (0\leq\beta<\alpha)$ such that
\begin{gather*}
|A_{0}(z_{j})|\geq\exp\{\alpha|z_{j}|^{\mu}\}, \\
|A_{n}(z_{j})|\leq\exp\{\beta|z_{j}|^{\mu}\} \quad (n=1,\dots,k-1),
\end{gather*}
as $j\to\infty$. Then every solution $f\not\equiv 0$ of
\eqref{e1.1} has infinite order.
\end{theorem}

Now there exists another question: For so many solutions of
infinite order, how to describe precisely the properties of growth
of solutions of infinite order of \eqref{e1.1}?

In this paper, we improve and extend Theorem \ref{thm1} by making use of
the concept of iterated order. Let us define inductively, for
$r\in [0,+\infty), \exp^{[1]}r=e^{r}$ and
$\exp^{[n+1]}r=\exp(\exp^{[n]}r), n\in \mathbb{N}$. For all $r$
sufficiently large, we define $\log^{[1]}r=\log r$ and
$\log^{[n+1]}r=$ $\log(\log^{[n]}r), n\in \mathbb{N}$. We also
denote $\exp^{[0]}r$$=r=$$\log^{[0]}r$, $\log^{[-1]}r=\exp^{[1]}r$
and $\exp^{[-1]}r$$=\log^{[1]}r$. We recall the following
definitions (see \cite{kinnunen,sato,cao-chen-zheng-tu}).

\begin{definition} \label{def1} \rm
The iterated $p$-order $\sigma _{p}(f)$ of a
meromorphic function $f(z)$ is defined by
$$
\sigma_{p}(f)=\limsup_{r\to \infty} \frac{\log^{[p]}T(r,f)}{\log r } \quad(p\in
\mathbb{N}).
$$
\end{definition}

\begin{remark} \label{rmk1} \rm
(1). If $p=1$, then we denote $\sigma _{1}(f)=\sigma(f)$;
(2). If $p=2$. then we denote by $\sigma_{2}(f)$ the so-called hyper
order (see \cite{yi-yang});
(3). If $f(z)$ is an entire function,
then
$$
\sigma _{p}(f)=\limsup_{r\to\infty} \frac{\log^{[p+1]}M(r,f)}{\log r }.
$$
\end{remark}

\begin{definition} \label{def2} \rm
The growth index of the iterated order of a
meromorphic function $f(z)$ is defined by
$$
i(f)=\begin{cases}
0 & \text{if $f$ is  rational},  \\
\min\{n\in \mathbb{N}:\sigma _{n}(f)<\infty\}
& \text{if $f$  is transcendental and}\\
& \text{$\sigma_{n}(f)<\infty $ for  some $n\in \mathbb{N}$}, \\
\infty & \text{if $\sigma _{n}(f)=\infty $ for all $n\in \mathbb{N}$.}
\end{cases}
$$
Similarly, we can define the iterated lower order $\mu_{p}(f)$
of a meromorphic function $f(z)$ and the growth index $i_{\mu}(f)$
of $\mu_{p}(f)$.
\end{definition}

\begin{definition} \label{def3} \rm
The iterated convergence exponent of the sequence of
$a$-points  $(a\in \mathbb{C}\cup\{\infty\})$ is defined by
$$
\lambda_{n}(f-a)=\lambda_{n}(f,a)=\limsup_{r\to \infty}
\frac{\log^{[n]}N(r,\frac{1}{f-a})}{\log r}\quad(n\in
\mathbb{N}),
$$
and $\overline{\lambda}_{n}(f-a)$, the  iterated
convergence exponent of the sequence of distinct $a$-points is
defined by
$$
\overline{\lambda}_{n}(f-a)=\overline{\lambda}_{n}(f,a)=
\limsup_{r\to\infty} \frac{\log^{[n]}\overline{N}(r,\frac{1}{f-a})}{\log
r}\quad(n\in \mathbb{N}).
$$
\end{definition}

\begin{remark} \label{rmk3} \rm
(1). $\lambda_{1}(f-a)=\lambda(f-a)$; (2).
$\overline{\lambda}_{1}(f-a)=\overline{\lambda}(f-a)$.
\end{remark}

\begin{definition} \label{def4} \rm
The growth index of the iterated convergence exponent of the
sequence of a-points of a meromorphic function $f(z)$ with iterated
order is defined by
$$
i_{\lambda}(f-a)=i_{\lambda}(f,a)
=\begin{cases}
0 & \text{if }n(r,\frac{1}{f-a})=O(\log r),  \\
\min\{n\in \mathbb{N}:\lambda _{n}(f)<\infty\}
&\text{if $\lambda_{n}(f-a)<\infty$ for }\\
&\text{some $n\in \mathbb{N}$}, \\
\infty &\text{if $\lambda _{n}(f-a)=\infty$ for }\\
&\text{all $n\in \mathbb{N}$}.
\end{cases}
$$
Similarly, we can define the growth index
$i_{\overline{\lambda}}(f-a)$ of
$\overline{\lambda}_{p}(f-a)$.
\end{definition}

Now we show one of our main result, which improves and extends
Theorem \ref{thm1}.

\begin{theorem}  \label{thm2}
Suppose there exist  a
sequence of points $(z_{j})_{j\in \mathbb{N}}$ with
$\lim_{j\to\infty}z_{j}=\infty$, and two real
numbers $\alpha, \beta$ ($0\leq\beta<\alpha$) such that for any
 $\varepsilon>0$,
\begin{equation}
|A_{0}(z_{j})|\geq\exp^{[p]}\{\alpha|z_{j}|^{\sigma-\varepsilon}\}\label{e1.2}
\end{equation}
and for $n=1,\dots,k-1$,
\begin{equation}
|A_{n}(z_{j})|\leq\exp^{[p]}\{\beta|z_{j}|^{\sigma-\varepsilon}\}
\quad (),\label{e1.3}
\end{equation}
as $j\to\infty$, where $p$ is a positive integer number,
and $\sigma$ satisfies
$\sigma_{p}(A_{n})\leq\sigma_{p}(A_{0})=\sigma\, $
$(n=1,2,\dots,k-1)$. Then every solution $f\not\equiv 0$ of
\eqref{e1.1} is of infinite order with $i(f)=p+1$ and
$\sigma_{p+1}(f)=\sigma_{p}(A_{0})$.
\end{theorem}

When $p=1$ in Theorem \ref{thm2}, we obtain  the following
corollary.

\begin{corollary} \label{coro1}
 Suppose  there exist  a
sequence of points $(z_{j})_{j\in \mathbb{N}}$ with
$\lim_{j\to\infty}z_{j}=\infty$, and two real
numbers $\alpha, \beta (0\leq\beta<\alpha)$ such that for any
given $\varepsilon>0$,
\[
|A_{0}(z_{j})|\geq\exp\{\alpha|z_{j}|^{\sigma-\varepsilon}\}
\]
and for $n=1,\dots,k-1$,
$$
|A_{n}(z_{j})|\leq\exp\{\beta|z_{j}|^{\sigma-\varepsilon}\}
$$
as $j\to\infty$, where $\sigma$ satisfies
$\sigma(A_{n})\leq\sigma(A_{0})=\sigma<\infty$$\,(n=1,2,\dots,k-1)$.
Then every solution $f\not\equiv 0$ of \eqref{e1.1} satisfies
$\sigma(f)=\infty$ and $\sigma_{2}(f)=\sigma(A_{0})$.
\end{corollary}

From Theorem \ref{thm2}, we can also deduce the following results.

\begin{corollary}[Kinnunen \cite{kinnunen}] \label{coro2}
  Let $A_{0}, A_{1}, \dots, A_{k-1} $ be
entire functions such that $i(A_{0})=p (0<p<\infty)$ and
$\sigma_{p}(A_{0})=\sigma$. If either $\max\{i(A_{j}):
j=1,\dots,k-1\}<p$ or $\max\{\sigma_{p}(A_{j}):
j=1,\dots,k-1\}<\sigma_{p}(A_{0})$. Then all solutions $f
\not\equiv 0$ of \eqref{e1.1} satisfy $i(f)=p+1$ and
$\sigma_{p+1}(f)=\sigma_{p}(A_{0})$.
\end{corollary}

\begin{corollary}[Chen, Yang \cite{chen-yang-1}] \label{coro3}
Let $A_{0}, A_{1}, \dots, A_{k-1} $
be entire functions satisfying
$$
\max\{\sigma(A_{j}),(j=1,\dots,k-1)\}<\sigma(A_{0})<\infty,
$$
then all solutions $f \not\equiv 0$ of \eqref{e1.1} satisfy
$\sigma_{2}(f)=\sigma(A_{0})$.
\end{corollary}

Considering nonhomogeneous linear differential equations
\begin{equation}
f^{(k)}+A_{k-1}f^{(k-1)}+\dots+A_{0}f=F\label{e1.4}
\end{equation}
corresponding to \eqref{e1.1},  we obtain
the following results of the iterated order and iterated
convergence exponent of zeros of solutions of
\eqref{e1.1}.

\begin{theorem} \label{thm3}
Assume that $A_{0},\dots,A_{k-1}$ satisfy the hypotheses of
Theorem \ref{thm2}. Let $F\not\equiv 0$ be an entire function with
$i(F)=q$, where $q\in \mathbb{N}$. Then
\begin{enumerate}
\item  If either $q>p+1$, or $q=p+1$ while
$\sigma_{q}(F)>\sigma_{p}(A_{0})$, then all solutions $f$ of
\eqref{e1.4} satisfy $i(f)=i(F)=q$ and
$\sigma_{q}(f)=\sigma_{q}(F).$\par

\item  If either $q<p+1$, or $q=p+1$ while
$\sigma_{q}(F)<\sigma_{p}(A_{0})$, then all solutions $f(z)$ of
\eqref{e1.4} satisfy
$i(f)=i_{\lambda}(f)=i_{\overline{\lambda}}(f)=p+1$ and
$\sigma_{p+1}(f)=\lambda_{p+1}(f)=\overline{\lambda}_{p+1}(f)=\sigma_{p}(A_{0})$,
with at most one exception.
\end{enumerate}
\end{theorem}

The methods and tactics in the proof of Theorem \ref{thm1} are mainly the
estimate for the logarithmic derivative of a transcendental
meromorphic function of finite order due to Gundersen (see
\cite{gundersen-1}). In this paper, To prove Theorem \ref{thm2}, on the one
hand, we mainly use the estimate for the logarithmic derivative  of
a transcendental meromorphic function of finite or infinite order
due to Gundersen (see \cite{gundersen-1}); on the other hand, we
make mainly use of the Winman-Valiron theory \cite{he-xiao}. We
prove Theorem \ref{thm3} by making use of  the basic method of the
nonhomogeneous linear differential equations,  Lemmas \ref{lem6} and
\ref{lem7} due to the present author and other three authors (see
\cite{cao-chen-zheng-tu}).

\section{Lemmas}

\begin{lemma}[Winman-Valiron \cite{he-xiao}] \label{lem1}
 Let $f(z)$ be a transcendental entire function,
$\delta$ is a constant such that $0<\delta<\frac{1}{8}$, and let
$z$ be a point with $|z|=r$ at which
$|f(z)|>M(r,f)\cdot \nu_{f}(r)^{-\frac{1}{8}+\delta}$,
where  $\nu_{f}(r)$ denote the central index of $f$, then the
estimation
\begin{equation}
\frac{f^{(n)}(z)}{f(z)}=\big(\frac{\nu_{f}(r)}{z}\big)^{n}
\big(1+\eta_{k}(z)\big)\quad(n\in \mathbb{N}),\label{e2.1}
\end{equation}
holds for all $|z|=r$ outside a subset $E$ of finite logarithmic measure,
where
$$
\eta_{k}(z)=O\Big((\nu_{f}(r))^{-\frac{1}{8}+\delta}\Big).
$$
\end{lemma}

\begin{lemma}[{Kinnunen \cite[Remark 1.3]{kinnunen}}] \label{lem2}
If $f$ is a meromorphic function
with $i(f)=p\geq 1$, then
$\sigma_{p}(f)=\sigma_{p}(f')$.
\end{lemma}

\begin{lemma}[Bank \cite{bank}] \label{lem3}
Let $g:[0,\infty)\to \mathbb{R}$
and $h:[0,\infty)\to \mathbb{R}$ be monotone nondecreasing
functions such that $g(r)\leq h(r)$ outside of an exceptional set
$E_{2}$ of finite linear measure. Then for any $\alpha>1$, there
exists  $r_{0}$ such that $g(r) \leq h(\alpha r)$ for all
$r>r_{0}$.
\end{lemma}

\begin{lemma}[Gundersen \cite{gundersen-1}] \label{lem4}
Let $f$ be a transcendental meromorphic function
of finite order $\sigma$. Let $\varepsilon>0$ be a constant, and $k$
and $j$ be integers satisfying $k>j\geq 0$. Then the following two
statements hold:
\begin{itemize}
\item[(a)] There exists a set $E_{1}\subset(1,\infty)$ which has finite
logarithmic measure, such that for all $z$ satisfying $|z|\not\in
E_{1}\bigcup[0,1]$, we have
\begin{equation}
|\frac{f^{(k)}(z)}{f^{(j)}(z)}|\leq|z|^{(k-j)(\sigma-1+\varepsilon)}.
\label{e2.2}
\end{equation}

\item[(b)] There exists a set $E_{2}\subset [0,2\pi)$ which has linear
measure zero, such that if $\theta\in[0,2\pi)-E_{2}$, then there
is a constant $R=R(\theta)>0$ such that \eqref{e2.2} holds for all
$z$ satisfying  $\arg z=\theta$ and $R\leq|z|$.
\end{itemize}
\end{lemma}

\begin{lemma}[Gundersen \cite{gundersen-1}] \label{lem5}
Let $f$ be a transcendental meromorphic function.
Let $\alpha>1$ be a constant,  and  $k$ and  $j$ be integers
satisfying $k> j\geq 0$. Then the following two statements hold:
\begin{itemize}
\item[(a)] There exists a set $E_{1}\subset(1,\infty)$ which has finite
logarithmic measure,  and a constant $C>0$, such that for all $z$
satisfying $|z|\not\in E_{1}\bigcup[0,1]$, we have (with $r=|z|$)
\begin{equation}
\big|\frac{f^{(k)}(z)}{f^{(j)}(z)}\big|
\leq C\big[\frac{T(\alpha r,f)}{r}(\log r)^{\alpha}\log T(\alpha r,
f)\big]^{k-j}.\label{e2.3}
\end{equation}

\item[(b)] There exists a set $E_{2}\subset [0,2\pi)$ which has linear
measure zero, such that if $\theta\in[0,2\pi)-E_{2}$, then there
is a constant $R=R(\theta)>0$ such that \eqref{e2.3} holds for all
$z$ satisfying  $\arg z=\theta$ and
$R\leq|z|$.
\end{itemize}
\end{lemma}

\begin{lemma}[Cao, Chen, Zheng and Tu \cite{cao-chen-zheng-tu}] \label{lem6}
 Let $g(z)$ be an entire function of finite iterated
order with $i(g)=m,i_{\mu}(g)=n,\sigma_{m}(g)=\sigma,\mu_{n}(g)=\mu$
(where $m,n\in \mathbb{N}$), and $\nu_{g}(r)$ denote the central index
of $g$, then
\begin{gather*}
\limsup_{r\to\infty} \frac{\log^{[m]}\nu_{g}(r)}{\log r}=\sigma_{m}(g)
=\sigma,\\
\liminf_{r\to\infty} \frac{\log^{[n]}\nu_{g}(r)}{\log
r}=\mu_{n}(g)=\mu.
\end{gather*}
\end{lemma}

\begin{lemma}[Cao, Chen, Zheng and Tu \cite{cao-chen-zheng-tu}]
\label{lem7}
 Let  $f(z)$ be a meromorphic solution of the differential equation
$$
f^{(k)}+B_{k-1}f^{(k-1)}+\dots+B_{0}f=F,
$$
where $B_{0},\dots,B_{k-1},F\not\equiv 0$ are meromorphic functions, such
that
\\
(1) $\max\{i(F),i(B_{j})(j=0,\dots,k-1)\}<i(f):=p$, $(0<p<\infty)$;
or \\
(2) $\max\{\sigma_{p}(F),\sigma_{p}(B_{j})(j=0,\dots,k-1)\}<\sigma_{p}(f)$.
Then $i_{\overline{\lambda}}(f)=i_{\lambda}(f)=i(f)=p$  and
$\overline{\lambda}_{p}(f)=\lambda_{p}(f)=\sigma_{p}(f)$.
\end{lemma}

\section{Proof of main results}

 \begin{proof}[Proof of Theorem \ref{thm2}]
Let $f\not\equiv 0$ be a solution of  \eqref{e1.1}.
Then we have
\begin{equation}
|A_{0}(z)|\leq|\frac{f^{(k)}(z)}{f(z)}|+|A_{k-1}(z)||\frac{f^{(k-1)}(z)}
{f(z)}|+\dots+|A_{1}(z)||\frac{f'(z)}{f(z)}|.\label{e3.1}
\end{equation}
When $p=1$, from Theorem \ref{thm1}, we have $\sigma(f)=\infty$. when
$p>1$, from Lemma \ref{lem4} (a) and using similar discussion as to the
proof of Theorem \ref{thm1} (see \cite[page 242-243]{belaidi-hamouda}), one
can also deduce that $\sigma(f)=\infty$.

By Lemma \ref{lem5} (a), There exist a set $E_{1}\subset(1,\infty)$ which
has finite logarithmic measure,  and  positive constants $c$ and
$M$ such that for all $z_{j}$ satisfying $|z_{j}|\not\in
E_{1}\bigcup[0,1]$, we have (with $r_{j}=|z_{j}|$),
\begin{equation}
|\frac{f^{(t)}(z_{j})}{f(z_{j})}|\leq
M\cdot\left[r^{c}_{j}\cdot
T(2r_{j},f)\right]^{2k}\quad(t=1,\dots,k-1). \label{e3.2}
\end{equation}
Substituting \eqref{e1.2}, \eqref{e1.3} and \eqref{e3.2} into
\eqref{e3.1}, we obtain
$$
\exp^{[p]}\{\alpha|z_{j}|^{\sigma-\varepsilon}\}\leq|A_{0}(z_{j})|
\leq M\cdot\left[r_{j}^{c}\cdot T(2r_{j},f)\right]^{2k} \cdot
k\cdot\exp^{[p]}\{\beta|z_{j}|^{\sigma-\varepsilon}\}
$$
as $z_{j}\to\infty$, $|z_{j}|\not\in E_{1}\bigcup[0,1]$.
Since $\alpha>\beta\geq 0$, we get that
$$
\exp\left((\alpha-\beta)|z_{j}|^{\sigma-\varepsilon}\right)
\leq M\left[r_{j}^{c}T(2r_{j},f)\right]^{2k} k,
$$
for $p=1$,
$$
\exp\left((1-\gamma)\exp^{[p-1]}\{\alpha|z_{j}|^{\sigma-\varepsilon}\}\right)
\leq M\left[r_{j}^{c} T(2r_{j},f)\right]^{2k} k,
$$
for $p>1$, as $z_{j}\to\infty$, $|z_{j}|\not\in E_{1}\bigcup[0,1]$,
where $\gamma$ $(0<\gamma<1)$ is a  real number. Hence from
Definition \ref{def1}, Definition \ref{def2} and Lemma \ref{lem3} , we can deduce that
$i(f)\geq p+1$ and
\begin{equation}
\sigma_{p+1}(f)+\varepsilon\geq\sigma=\sigma_{p}(A_{0}).\label{e3.3}
\end{equation}

On the other hand, by Lemma \ref{lem1}, there exists a set
$E_{2}\subset\{1,+\infty\}$ with finite logarithmic measure, when
$|z|=r\not\in[0,1]\bigcup E_{2}$, and $|f(z)|=M(r,f)$, we have
\begin{equation}
\frac{f^{(m)}(z)}{f(z)}=\big(\frac{\nu_{f}(r)}{z}\big)^{m}(1+o(1))
\quad(m=1,\dots,k).\label{e3.4}
\end{equation}
Since $\sigma=\sigma_{p}(A_{0})\geq\sigma_{p}(A_{n}),
(n=1,2,\dots,k-1)$, then for any given $\varepsilon(>0)$ and
sufficiently large $r$, and by Definition \ref{def1}, we get
\begin{equation}
|A_{i}(z)|\leq \exp^{[p]}\{r^{\sigma+\varepsilon}\}\quad(i=0,\dots,k-1).
\label{e3.5}
\end{equation}
Substituting \eqref{e3.4} and \eqref{e3.5} into  \eqref{e1.1},
we get
\begin{equation}
\begin{aligned}
&\big(\frac{\nu_{f}(r)}{|z|}\big)^{k}|1+o(1)| \\
&\leq\big(\frac{\nu_{f}(r)}{|z|}\big)^{k-1}|1+o(1)|(|A_{k-1}|+\dots+|A_{0}|)
\\
&\leq k\big(\frac{\nu_{f}(r)}{|z|}\big)^{k-1}|1+o(1)|\exp^{[p]}
\{r^{\sigma+\varepsilon}\}, \quad
r\not\in[0,1]\bigcup E_{2}.
\end{aligned}\label{e3.6}
\end{equation}
By Lemma \ref{lem3}, Lemma \ref{lem6} and \eqref{e3.6}, we obtain $i(f)\leq p+1$ and
\begin{equation}
\sigma_{p+1}(f)=\limsup_{r\to\infty} \frac{\log^{[p+1]}\nu_{f}(r)}{\log r}
\leq\sigma+\varepsilon.\label{e3.7}
\end{equation}
Since $\varepsilon$ is arbitrary, we get from \eqref{e3.3} and \eqref{e3.7}
that Theorem \ref{thm2} holds.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
We can assume that $\{f_{1},\dots,f_{k}\}$ is an entire solution
base of \eqref{e1.1}. By Theorem \ref{thm2}, we know that $i(f_{j})=p+1$
and $\sigma_{p+1}(f_{j})=\sigma_{p}(A_{0})$. Thus any solution of
\eqref{e1.4} has the form
\begin{equation}
f(z)=B_{1}f_{1}+B_{2}f_{2}+\dots+B_{k}f_{k},\label{e4.1}
\end{equation}
where $B_{1},\dots,B_{k}$ are suitable entire functions satisfying
\begin{equation}
B_{j}'=F\cdot G_{j}(f_{1},\dots,f_{k})\cdot W(f_{1},\dots,f_{k})^{-1}\quad
 (j=1,\dots,k),\label{e4.2}
\end{equation}
where $G_{j}$ $(f_{1}$, $\dots,$$f_{k})$ is differential
polynomials in $f_{1}$, $\dots,f_{k}$ and their derivatives, and
$W(f_{1}$, $\dots$, $f_{k})$ is the Wronskian of
$f_{1},\dots,f_{k}$. By Lemma \ref{lem2} and the above-mentioned, we
obtain
\begin{equation}
i(f)\leq \max\{p+1,q\}.\label{e4.3}
\end{equation}

(1) If either $q>p+1$, or  $q=p+1$ while
$\sigma_{q}(F)>\sigma_{p}(A_{0})$, it  follows  from
\eqref{e4.1}-\eqref{e4.3} and Eq.\eqref{e1.4} that $i(f)=i(F)=q$
and $\sigma_{q}(f)=\sigma_{q}(F)$.

(2) If either $q<p+1$,or $q=p+1$ while
$\sigma_{q}(F)<\sigma_{p}(A_{0})$, it follows from \eqref{e4.1}
-\eqref{e4.3} and Eq.\eqref{e1.4} that either $i(f)<p+1$, or
$i(f)=p+1$ and $\sigma_{p+1}(f)\leq\sigma_{p}(A_{0})$.

Now we assert that all solutions $f$ of \eqref{e1.4} satisfy
$i(f)=p+1$ and $\sigma_{p+1}(f)=\sigma_{p}(A_{0})$ with at most
one exception. In fact, if there exist two entire solutions
$g_{1},g_{2}$ of \eqref{e1.4} which satisfy  either $i(g_{j})<p+1$
or $\sigma_{p+1}(g_{j})<\sigma_{p}(A_{0}),(j=1,2)$, then
$g=g_{1}-g_{2}$ is a solution of \eqref{e1.1} which satisfies
either $i(g)=i(g_{1}-g_{2})<p+1$ or
$\sigma_{p+1}(g)=\sigma_{p+1}(g_{1}-g_{2})<\sigma_{p}(A_{0})$. But
$g=g_{1}-g_{2}$ is a solution of \eqref{e1.1} satisfying
$i(g)=i(g_{1}-g_{2})=p+1$ and
$\sigma_{p+1}(g)=\sigma_{p+1}(g_{1}-g_{2})=\sigma_{p}(A_{0})$ by
Theorem \ref{thm2}. This is a contradiction.

By Lemma \ref{lem7}, we know that all solutions $f$ of \eqref{e1.4} with
$i(f)=p+1$ and $\sigma_{p+1}(f)=\sigma_{p}(A_{0})$ satisfy
$i_{\overline{\lambda}}(f)=i_{\lambda}(f)=i(f)=p+1$ and
$\sigma_{p+1}(f)=\overline{\lambda}_{p+1}(f)=\lambda_{p+1}(f)$.
Therefore, we deduce that Theorem \ref{thm3} holds.
\end{proof}

\begin{proof}[Proofs of  Corollaries \ref{coro2} and \ref{coro3}]
Let $\max\limits_{1\leq n\leq
k-1}\{\sigma_{p}(A_{n})\}=b<\sigma_{p}(A_{0})=\sigma$. Then for
any given $\varepsilon$, $0<\varepsilon<(\sigma-b)/2$, by
Definition \ref{def1}, we have
\begin{gather*}
|A_{0}(z)|>\exp^{[p]}\{|z|^{\sigma-\varepsilon}\}, \\
|A_{n}(z)|<\exp^{[p]}\{|z|^{b+\varepsilon}\}
\leq\exp^{[p]}\{\delta|z|^{\sigma-\varepsilon}\}\quad
(n=1,\dots,k-1),
\end{gather*}
 for sufficiently large $|z|$, where $\delta$
$(0<\delta<1)$ is a real number. By making use of the above two
inequalities and Theorem \ref{thm2}, we get that Corollary \ref{coro2}
follows.
Corollary \ref{coro3} is just a special case of Corollary \ref{coro2} when $p=1$
\end{proof}

\subsection*{Acknowledgements}
The author is grateful to Professor Hong-Xun
Yi for his inspiring guidance, and  to  the anonymous
referee for his/her valuable suggestions and improvements to the
present paper.

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\end{document}