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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 195, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/195\hfil Meromorphic coefficients of (p,q)-order]
{Solutions for linear differential equations \\ with meromorphic coefficients
of (p,q)-order\\ in the plane}

\author[L.-M. Li, T.-B. Cao \hfil EJDE-2012/195\hfilneg]
{Lei-Min Li, Ting-Bin Cao}  % in alphabetical order

\address{Lei-Min Li \newline
Department of Mathematics, Nanchang University, Nanchang,
Jiangxi 330031, China}
\email{leiminli@hotmail.com}

\address{Ting-Bin Cao \newline
Department of Mathematics, Nanchang University, Nanchang,
Jiangxi 330031, China}
\email{tbcao@ncu.edu.cn, tingbincao@hotmail.com (corresponding author)}

\thanks{Submitted January 6, 2012. Published November 8, 2012.}
\subjclass[2000]{34M10, 30D35, 34M05}
\keywords{Linear differential equation; meromorphic function;
 (p,q)-order;\hfill\break\indent  Nevanlinna theory}

\begin{abstract}
 In this article we study the growth of meromorphic solutions
 of high order linear differential equations  with meromorphic
 coefficients of $(p,q)$-order.  We  extend some previous
 results due to  Bela\"{\i}di, Cao-Xu-Chen,  Kinnunen, Liu- Tu -Shi,
 and others.
\end{abstract}

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

\section{Introduction and main results}

For $k\geq 2$, consider the linear differential equations
\begin{gather}\label{E-1}
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots+A_1(z)f'+A_0(z)f=0,\\ 
\label{E-2}
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots+A_1(z)f'+A_0(z)f=F(z),
\end{gather}
 where $A_0\not\equiv 0$ and $F\not\equiv 0$.
When the coefficients $A_0, A_1, \dots, A_{k-1}$ and $F$ are
entire functions, it is well known that all solutions of \eqref{E-1}
and \eqref{E-2} are entire functions, and that if some coefficients
of \eqref{E-1} are transcendental then \eqref{E-1} has at least one
solution with infinite order. We refer to \cite{12} for the
literature on the growth of entire solutions of \eqref{E-1} and
\eqref{E-2}.

As far as we known, Bernal \cite{1} firstly introduced the idea of
iterated order to express the fast growth of solutions of complex
linear differential equations. Since then, many authors obtained
further results on iterated order of solutions of \eqref{E-1} and
\eqref{E-2}, see e.g. \cite{belaidi-1, belaidi-2, 1, 3, 5, 11, 15}.
Recently, Liu, Tu and Shi \cite{13} firstly introduced the concept
of (p, q)-order for the case $p\geq q\geq 1$ to investigate the
entire solutions of \eqref{E-1} and \eqref{E-2}, and obtained some
results which improve and generalize some previous results.

\begin{theorem}[{\cite[Theorems 2.2-2.3]{13}}] \label{T-A} 
Let $p\geq q\geq 1$, and let
$A_0, A_1, \dots, A_{k-1}$ be entire functions such that either
$$
\max\{\sigma_{(p,q)}(A_j): j \neq 0\} < \sigma_{(p,q)}(A_0) <
+\infty,
$$ 
or
\begin{gather*}
\max\{\sigma_{(p,q)}(A_j): j \neq 0\} \leq \sigma_{(p,q)}(A_0) <
+\infty,\\
 \max\{\tau_{(p,q)}(A_j): \sigma_{(p,q)}(A_j) = \sigma_{(p,q)}(A_0)
> 0\} < \tau_{(p,q)}(A_0),
\end{gather*}
then every nontrivial solution $f$ of
\eqref{E-1} satisfies $\sigma_{(p+1,q)}(f) =
\sigma_{(p,q)}(A_0)$.
\end{theorem}

Recently, Cao, Xu and Chen \cite{3} considered the growth of
meromorphic solutions of equations \eqref{E-1} and \eqref{E-2} with
meromorphic coefficients of finite iterated order,  and obtained
some results which improve and generalize some previous results.

\begin{theorem}[{\cite[Theorem 2.1]{3}}] \label{T-B}
Let $A_0, A_1, \dots, A_{k-1}$ be meromorphic functions in the
plane, and let $i(A_0)=p$ $(0<p<\infty)$. Assume that either
$i_{\lambda}(\frac{1}{A_0})<p$ or
$\lambda_p(\frac{1}{A_0})<\sigma_p(A_0)$, and that either
$$
\max\{i(A_j): j=1,2, \dots,k-1\}<p
$$ 
or
\begin{gather*}
\max\{\sigma_p(A_j): j=1,2,\dots,k-1\}\leq\sigma_p(A_0):=\sigma \quad
 (0<\sigma<\infty),\\
\max\{\tau_p(A_j): \sigma_p(A_j)=\sigma_p(A_0)\}<\tau_p(A_0):=\tau\quad
 (0<\tau<\infty).
\end{gather*}
Then every meromorphic solution $f\not\equiv 0$ whose poles are of
uniformly bounded multiplicities, of equation \eqref{E-1} satisfies
$i(f)=p+1$ and $\sigma_{p+1}(f)=\sigma_p(A_0)$.
\end{theorem}

There exists a natural question:
\emph{How about the growth of
meromorphic solutions of equations \eqref{E-1} and \eqref{E-2} with
meromorphic coefficients of finite (p, q)-order in the
plane?}

The main purpose of this paper is to consider the above question.
Now we show our main results. For homogeneous linear differential
equation \eqref{E-1}, we obtain the following results. 

\begin{theorem} \label{T-1} 
Let $A_0, A_1,\dots,A_{k-1}$ be meromorphic
functions in the plane. Suppose that there exists one coefficient
$A_s$ $(s\in\{0,1,\dots,k-1\})$ such that
\[
\max\{\sigma_{(p,q)}(A_j), \lambda_{(p,q)}(\frac{1}{A_s}):j\neq
s\}<\sigma_{(p,q)}(A_s)<+\infty,
\]
then every transcendental meromorphic solution $f$ whose poles are
of uniformly bounded multiplicities of \eqref{E-1} satisfies
\[
\sigma_{(p+1,q)}(f)\leq \sigma_{(p,q)}(A_s)\leq \sigma_{(p,q)}(f).
\]
Furthermore, if all solutions of \eqref{E-1} are meromorphic solutions, then
there is at least one meromorphic solution, say $f_1$, satisfies
\[
\sigma_{(p+1,q)}(f_1)=\sigma_{(p,q)}(A_s).
\]
\end{theorem}

Now replacing the arbitrary coefficient $A_s$ by the dominant fixed 
coefficient $A_0$, then we obtain the following result.

\begin{theorem}\label{T-2}
Let $A_0,A_1,\dots,A_{k-1}$ be meromorphic functions in the
plane satisfying
\[
\max\{\sigma_{(p,q)}(A_j),
\lambda_{(p,q)}(\frac{1}{A_0}):j=1,2,\dots,k-1\}<\sigma_{(p,q)}(A_0)<+\infty,
\]
then every meromorphic solution $f$ whose poles are of uniformly
bounded multiplicities of \eqref{E-1} satisfies
\[
\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0).
\]
\end{theorem}

If there exist some other coefficients $A_j$
$(j\in\{1,2,\dots,k-1\})$ having the same (p,q)-order as $A_0$,
then we have the following result by making use of the concept of
(p,q)-type.

\begin{theorem}\label{T-3}
Let $A_0,A_1,\dots,A_{k-1}$ be meromorphic functions in the
plane, assume that
\[
\lambda_{(p,q)}(\frac{1}{A_0})<\sigma_{(p,q)}(A_0)
\]
and
\begin{gather*}
\max\{\sigma_{(p,q)}(A_j):j=1,2,\dots,k-1\}=
\sigma_{(p,q)}(A_0)<+\infty,
\\
\max\{\tau_{(p,q)}(A_j):\sigma_{(p,q)}(A_j)=\sigma_{(p,q)}(A_0)>0\}
<\tau_{(p,q)}(A_0).
\end{gather*} 
Then any nonzero meromorphic solution $f$ whose poles are of uniformly
bounded multiplicities of \eqref{E-1} satisfies
\[
\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0).
\]
\end{theorem}

Obviously, Theorems \ref{T-2} and \ref{T-3} are a generalization of
Theorems \ref{T-A} and \ref{T-B}.
Considering nonhomogeneous linear differential equation \eqref{E-2}, we obtain
the following three results.

\begin{theorem}\label{T-4}
Assume that $A_0,A_1,\dots,A_{k-1},F\not\equiv 0$ be meromorphic
functions in the plane satisfying
\[
\max\{\sigma_{(p,q)}(A_j), \lambda_{(p,q)}(\frac{1}{A_0}),
\sigma_{(p+1,q)}(F):j=1,2,\dots,k-1\}<\sigma_{(p,q)}(A_0),
\]
then all meromorphic solutions $f$ whose poles are of uniformly
bounded multiplicities of \eqref{E-2} satisfy
\[
\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)
=\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0)
\]
with at most one exceptional solution $f_0$ satisfying
$\sigma_{(p+1,q)}(f_0)<\sigma_{(p,q)}(A_0)$.
\end{theorem}

\begin{theorem}\label{T-5}
Let $A_0,A_1,\dots,A_{k-1},F\not\equiv 0$ be meromorphic functions
in the plane satisfying
\[
\max\{\sigma_{(p,q)}(A_j):j=0,1,\dots,k-1\}<\sigma_{(p+1,q)}(F).
\]
Suppose that all solutions of \eqref{E-2} are meromorphic functions whose
poles are of uniformly bounded multiplicities, then
$\sigma_{(p+1,q)}(f)=\sigma_{(p+1,q)}(F)$ holds for all
solutions of \eqref{E-2}.
\end{theorem}


\begin{theorem}\label{T-6}
Let $H\subset (1,\infty)$ be a set satisfying
$\overline{\log dens}\{|z|: |z|\in H\}>0$ and let $A_0,A_1,\dots$,
$A_{k-1}$, $F\not\equiv 0$ be meromorphic functions in the plane satisfying
\[\max\{\sigma_{(p,q)}(A_j):j=1,2,\dots,k-1\}<\alpha_1,
\]
where $\alpha_1$ is a constant, and there exists another constant
$\alpha_2$ ($\alpha_2<\alpha_1$) such than for any given
$\epsilon$ ($0<\epsilon<\alpha_1-\alpha_2$), we have
\[
|A_0{(z)}|\geq\exp_{p+1}\{(\alpha_1-\epsilon)\log_qr\},
|A_j{(z)}|\leq\exp_{p+1}\{\alpha_2\log_qr\}
\]
for $|z|\in H$, $j=1,2,\dots,k-1$.
Then we have:

(i) If $\sigma_{(p+1,q)}(F)\geq \alpha_1$, then all meromorphic
solutions whose poles are of uniformly
bounded multiplicities of \eqref{E-2} satisfy
\[
\sigma_{(p+1,q)}(f)=\sigma_{(p+1,q)}(F).
\]

(ii) If $\sigma_{(p+1,q)}(F)< \alpha_1$,then all meromorphic
solutions whose poles are of uniformly
bounded multiplicities of \eqref{E-2} satisfy
\[\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f)=\alpha_1
\]
with at most one exceptional solution $f_2$ satisfying
\[\sigma_{(p+1,q)}(f_2)<\alpha_1.
\]
\end{theorem}

Recently, B. Bela\"{\i}di \cite{belaidi-3} investigated the growth of 
solutions of differential equations \eqref{E-1} and \eqref{E-2} with 
analytic coefficients of $(p,q)$-order
in the unit disc. So, it is also interesting to consider the growth 
of meromorphic solutions of differential equations with coefficients of 
$(p,q)$-order in the unit disc?

\section{Preliminaries and some lemmas}

We shall introduce some notation. Let us define inductively, for
$r\in [0,+\infty)$, $\exp_1r=e^{r}$ and
$\exp_{n+1}r=\exp(\exp_nr)$, $n\in\mathbb{N}$. For all $r$
sufficiently large, we define 
$\log_1r=\log^{+}r=\max\{\log r,0\}$ and 
$\log_{n+1}r=\log(\log_nr)$, $n\in\mathbb{N}$. We also
denote $\exp_0r=r=\log_0r$, $\log_{-1}r=\exp_1r$ and
$\exp_{-1}r=\log_1r$. Moreover, we denote the linear measure and
the logarithmic measure of a set $E\subset (1,\infty)$ by
$mE=\int_{E}dt$ and $m_{l}E=\int_{E}\frac{dt}{t}$. The upper
logarithmic density of $E\subset (1,\infty)$  is defined
 by
\[
\overline{\log {\rm dens}}{E}=\limsup_{r\to \infty}
\frac{m_{l}(E\cap\mathcal[1,r])}{\log r }.
\]

We assume that the reader is familiar with the fundamental results
and the standard notations of the Nevanlinna's value distribution
theory of meromorphic functions (e.g. see \cite{8,17}), such as
$T(r,f)$, $m(r,f)$, and $N(r,f)$. In this section, a meromorphic
function $f$ means meromorphic in the complex plane $\mathbb{C}$. To
express the rate of fast growth of meromorphic functions, we recall
the following definitions (e.g. see \cite{1,3,11,12,14}).

\begin{definition}\label{D-2.1} \rm
 The iterated $p$-order $\sigma_p(f)$ of a
meromorphic function $f$ is defined by 
\begin{equation*}
\sigma_p(f)=\limsup_{r\to \infty}\frac{\log_pT(r,f)}{\log r }
\quad(p\in\mathbb{N}).
\end{equation*}
If $f$ is
an entire function, then 
\begin{equation*}
\sigma _{p,M}(f)=\limsup_{r\to \infty}\frac{\log_{p+1}M(r,f)}{\log r
}\quad(p\in\mathbb{N}).
\end{equation*}
\end{definition}


\begin{definition} \rm
The growth index of the iterated order of a
meromorphic function $f$ is defined by
\[
i(f)=\begin{cases} 0 & \text{if $f$  is rational}, \\
\min\{n\in\mathbb{N}:\sigma_n(f)<\infty\}
& \text{if $f$ is transendental and $\sigma_n(f)<\infty$}\\
&\text{for some  $n\in\mathbb{N}$}, \\
\infty & \text{if $\sigma_n(f)=\infty$ for all $ n\in\mathbb{N}$.}
\end{cases}
\]
\end{definition}

\begin{definition} \rm
The iterated $p$-type of a
meromorphic function $f$ with iterated order  $p$-order
$0<\sigma_p(f)<\infty$ is defined by
\begin{equation*}
\tau_p(f)=\limsup_{r\to \infty}\frac{\log_{p-1}T(r,f)}{r^{\sigma_p(f)}}
\quad(p\in\mathbb{N}).
\end{equation*}
If $f$ is an entire function, then
\begin{equation*}
\tau_{p,M}(f)=\limsup_{r\to \infty}\frac{\log_pM(r,f)}{\
r^{\sigma_p(f)}} \quad(p\in\mathbb{N}).
\end{equation*}
\end{definition}


\begin{definition} \rm
The iterated convergence exponent of the
sequence of zeros of a meromorphic function $f$ is defined by
$$
\lambda_p(f)=\limsup_{r\to \infty}
\frac{\log_pN(r,\frac{1}{f})}{\log r}\quad(p\in\mathbb{N}).
$$
\end{definition}

\begin{definition} \rm
The growth index of the iterated convergence exponent of the
sequence of zeros of a meromorphic function $f$ with iterated order
is defined by
\[
i_{\lambda}(f)=\begin{cases}
0 & \text{if } n(r,\frac{1}{f})=O(\log r),  \\
\min\{n\in\mathbb{N}:\lambda _n(f)<\infty\} &
\text{if $\lambda_n(f)<\infty$ for some $n\in\mathbb{N}$,} \\
\infty & \text{if $\lambda _n(f)=\infty$ for all $n\in\mathbb{N}$.}
\end{cases} 
\]
\end{definition}

Similarly, we can use the notation $\overline{\lambda}_p(f)$
to denote the iterated convergence exponent of the sequence of
distinct zeros, and use the notation $i_{\overline{\lambda}}(f)$ to
denote the growth index of $\overline{\lambda}_p(f)$.

Now, we shall introduce the definition of meromorphic functions of
$(p,q)$-order, where $p$, $q$ are positive integers satisfying
$p\geq q\geq 1$. In order to keep accordance with Definition
\ref{D-2.1}, we will give a minor modification to the original
definition of $(p,q)$-order (e.g. see \cite{9,10}).

\begin{definition}\label{D-2.6} \rm
The $(p,q)$-order of a transcendental meromorphic function $f$ is defined 
by 
\begin{equation*}
\sigma_{(p,q)}(f)=\limsup_{r\to
\infty}\frac{\log_pT(r,f)}{\log_qr}.
\end{equation*}
If $f$ is a transcendental entire function, then
\begin{equation*}
\sigma _{(p,q)}(f)=\limsup_{r\to \infty}\frac{\log_{p+1}M(r,f)}{\log_qr }.
\end{equation*}
\end{definition}

It is easy to show that $0\leq\sigma _{(p,q)}\leq \infty$. By
Definition \ref{D-2.6} we note that
$\sigma_{(1,1)}(f)=\sigma_1(f)=\sigma(f)$,
$\sigma_{(2,1)}(f)=\sigma_2(f)$ and
$\sigma_{(p,1)}(f)=\sigma_p(f)$.

\begin{remark} \rm
 If $f$ is a meromorphic function satisfying $0\leq\sigma _{(p,q)}\leq
\infty$, then

(i)  $\sigma _{(p-n,q)}= \infty$ $(n<p)$, $\sigma _{(p,q-n)}=0$
 $(n<q)$, and $\sigma _{(p+n,q+n)}=1$ $(n<p)$ for $n=1$ to $\infty$.

(ii) If $(p_1,q_1)$ is another pair of integers satisfying
$p_1-q_1=p-q$ and $p_1<p$, then we have $\sigma
_{(p_1,q_1)}=0$ if $0<\sigma _{(p,q)}<1$ and $\sigma
_{(p_1,q_1)}=\infty$ if $1<\sigma _{(p,q)}<\infty$.

(iii) $\sigma _{(p_1,q_1)}=\infty$ for $ p_1-q_1>p-q$ and
 $\sigma _{(p_1,q_1)}=0$ for $p_1-q_1>p-q$.
\end{remark}

\begin{remark}\rm
Suppose that $f_1$ is a meromorphic function of
$(p,q)$-order $\sigma _1$ and $f_2$ is a meromorphic function of
$(p_1,q_1)$-order $\sigma _2$, let $p\leq p_1$. We can
easily deduce the result about their comparative growth:


(i) If $p_1-q_1>p-q$, then the growth of $f_1$ is slower than the
growth of $f_2$.

(ii) If $p_1-q_1<p-q$, then $f_1$ grows faster than $f_2$.

(iii) If $p_1-q_1=p-q>0$, then the growth of $f_1$ is slower than
the growth of $f_2$ if $\sigma _2\geq 1$,and the growth of $f_1$
is faster than the growth of $f_2$ if $\sigma _2< 1$.

(iv) Especially, when $p_1=p$ and $q_1=q$ then $f_1$ and $f_2$
are of the same index-pair $(p,q)$. If $\sigma _1>\sigma _2$,
then $f_1$ grows faster than $f_2$; and if $\sigma _1<\sigma_2$,
 then $f_1$ grows slower than $f_2$. If $\sigma _1=\sigma_2$, 
Definition 1.6 does not show any precise estimate about the
relative growth of $f_1$ and $f_2$.
\end{remark}

\begin{definition}\label{D-2.7} \rm
The $(p,q)$-type of a meromorphic function $f$ with $(p,q)$-order
$\sigma_{(p,q)}(f)\in(0, \infty)$ is defined by
\begin{equation*}
\tau_{(p,q)}(f)=\limsup_{r\to
\infty}\frac{\log_{p-1}T(r,f)}{(\log
_{q-1}r)^{\sigma_{(p,q)}(f)}}.
\end{equation*}
\end{definition}


\begin{definition} \rm
The $(p,q)$ convergence exponent of the
sequence of zeros of a meromorphic function $f$ is defined by
$$
\lambda_{(p,q)}(f)=\limsup_{r\to \infty}
\frac{\log_pN(r,\frac{1}{f})}{\log_qr}.
$$
\end{definition}

Similarly, we can use the notation $\overline{\lambda}_{(p,q)}(f)$
to denote the $(p,q)$ convergence exponent of the sequence of
distinct zeros of $f$.
To prove our results, we need the following lemmas.

\begin{lemma}[\cite{6}]\label{L-1}
Let $f_1,f_2,\dots,f_{k}$ be linearly
independent meromorphic solutions of the differential equation \eqref{E-1}
with meromorphic functions $A_0,A_1,\dots,A_{k-1}$ as the
coefficients, then
\[
m(r,A_j)=O\{\log(\max_{1\leq n\leq k}T(r,f_n))\}\quad (j=0,1,\dots,k-1).
\]
\end{lemma}

\begin{lemma}[\cite{18}] \label{L-2} 
Let $f$ be a meromorphic solution of equation \eqref{E-1}, assuming that not
all coefficients $A_j$ are constants. Given a real constant
$\gamma>1$, and denoting $T(r)=\Sigma^{k-1}_{j=0}T(r,A_j)$, we
have
\begin{gather*}
\log m(r,f)<T(r)\{(\log r)\log T(r)\}^{\gamma},\quad\text{if }s=0,\\ 
\log m(r,f)<r^{2s+\gamma-1}T(r)\{\log T(r)\}^{\gamma},\quad\text{if } s>0
\end{gather*}
outside of an exceptional set $E_s$ with
$\int_{E_s}t^{s-1}dt<\infty$.
\end{lemma}

By inequalities in \cite[Chapter 6]{23} and in 
\cite[Corollary 2.3.5]{12}, we obtain the following lemma.

\begin{lemma}\label{L-3}
If  $f$ is a meromorphic function, then
\[
\sigma_{(p.q)}(f)=\sigma_{(p.q)}(f').
\]
\end{lemma}

\begin{lemma}[\cite{7}] \label{L-4}
Let $f$ be a transcendental meromorphic function,
and let $\alpha$ be a given constant. Then there exist a set
$E_1\subset (1,\infty)$ that has finite logarithmic measure and a
constant $B>0$ depending only on $\alpha$ and
$(m,n)(m,n\in\{0,1,\dots ,k\})$, $m<n$ such that for all $z$ with
$|z|=r\not\in[0,1]\cup E_1$, we have
\[
\big|\frac{f^{(n)}(z)}{f^{(m)}(z)}\big|\leq B\Big(\frac{T(\alpha
r,f)}{r}(\log^{\alpha}r)T(\alpha r,f)\Big)^{n-m}.
\]
\end{lemma}

\begin{lemma}\label{L-5}
Let $f$ be a meromorphic function of $(p.q)$-order satisfying
$\sigma_{(p.q)}(f)<\infty$. Then there exists a set
 $E_2\subset (1,\infty)$ having infinite logarithmic measure such 
that for all $r \in E_2$, we have
\[
\lim_{r\to \infty}\frac{\log_pT(r,f)}{\log_qr
}=\sigma_{(p.q)}(f).
\]
\end{lemma}

\begin{proof}
By Definition \ref{D-2.6}, there exists a sequence
$\{r_n\}_{n=1}^{\infty}$ tending to $\infty$, satisfying
$(1+\frac{1}{n})r_n<r_{n+1}$, and
\[
\lim_{n\to \infty}\frac{\log_pT(r_n,f)}{\log_qr_n
}=\sigma_{(p.q)}(f).
\]
There exists a $n_1\in \mathbb{N}$, such that for $n\geq n_1$, and for
any $r \in [r_n,(1+\frac{1}{n})r_n]$, we have
\[
\frac{\log_pT(r_n,f)}{\log_q(1+\frac{1}{n})r_n } \leq
\frac{\log_pT(r,f)}{\log_qr } \leq
\frac{\log_pT((1+\frac{1}{n})r_n,f)}{\log_qr_n }.
\]
Set $E_2=\cup_{n=n_1}^{\infty}[r_n,(1+\frac{1}{n})r_n]$,
then for any $r \in E_2$, we have
\[
\lim_{r\to \infty}\frac{\log_pT(r,f)}{\log_qr
}=\lim_{n\to \infty}\frac{\log_pT(r_n,f)}{\log_qr_n
}=\sigma_{(p.q)}(f),
\]
where
\[
m_{l}E_2=\Sigma_{n=n_1}^{\infty}\int_{r_n}^{(1+\frac{1}{n})r_n}\frac{dt}{t}
=\Sigma_{n=n_1}^{\infty}\log (1+\frac{1}{n})=\infty. \qedhere
\]
\end{proof}

\begin{lemma}\label{L-6}
Let $\varphi(r)$ be a continuous and positive increasing
function, defined for $r \in [0,\infty]$ with
$\sigma_{(p.q)}(\varphi)=\limsup_{r\to \infty}
\frac{\log_p\varphi(r)}{\log_qr }$, then for any subset
$E_{3}\subset (0,\infty)$ that has a finite linear measure, there
exists a sequence $\{r_n\},r_n\not\in E_{3}$ such that
\[
\sigma_{(p.q)}(\varphi)=\lim_{r_n\to
\infty}\frac{\log_p\varphi(r_n)}{\log_qr_n }.
\]
\end{lemma}

\begin{proof}
Since $\sigma_{(p.q)}(\varphi)=\limsup_{r\to
\infty}\frac{\log_p\varphi(r)}{\log_qr }$, then there exists a
sequence $\{r_n'\}$ tending to $\infty$,such that
\[
\lim_{r_n'\to \infty}\frac{\log_p\varphi(r_n')}{\log_qr_n'
}=\sigma_{(p.q)}(\varphi).
\]
Set $mE_{3}=\delta < \infty$,then for $r_n \in
[r_n',r_n'+\delta+1]$, we have
\[
\frac{\log_p\varphi(r_n)}{\log_qr_n }\geq
\frac{\log_p\varphi(r_n')}{\log_q(r_n'+\delta+1) }
=\frac{\log_p\varphi(r_n')}{\log_{q-1}(\log{r_n'}+\log{(1+\frac{\delta+1}{r_n'})})
}.
\]
Hence
\begin{align*}
\lim_{r_n\to \infty}\frac{\log_p\varphi(r_n)}{\log_qr_n }
&\geq \lim_{r_n'\to \infty}\frac{\log_p\varphi(r_n')}
 {\log_{q-1}(\log{r_n'}+\log{(1+\frac{\delta+1}{r_n'})})} \\
&=\lim_{r_n'\to \infty}\frac{\log_p\varphi(r_n')}{\log_qr_n'}
 =\sigma_{(p.q)}(\varphi),
\end{align*}
this gives
\[
\sigma_{(p.q)}(\varphi)=\lim_{r_n\to
\infty}\frac{\log_p\varphi(r_n)}{\log_qr_n }. \qedhere
\]
\end{proof}

\begin{lemma}[\cite{9}]\label{L-7}
Let $f$ be an entire function of $(p.q)$-order,and
let $\nu_{f}(r)$ be the central index of $f$, then
\[
\limsup_{r\to \infty}\frac{\log_p\nu_{f}(r)}{\log_qr
}=\sigma_{(p.q)}(f).
\]
\end{lemma}


\begin{lemma}\label{L-8}
Let $f$ be a meromorphic function of $(p.q)$-order satisfying
$0<\sigma_{(p.q)}(f)<\infty$, let $\tau_{(p.q)}(f)>0$, then for
any given $\tau_{(p.q)}(f)>\beta$, there exists a set $E_4\subset
(1,\infty)$ that has infinite logarithmic measure such that for
all $r \in E_4$, we have
\[
\log_{p-1}T(r,f)>\beta(\log_{q-1}{r})^{\sigma_{(p.q)}(f)}.
\].
\end{lemma}

\begin{proof}
(i) (see \cite{3}) when $q=1$, it holds absolutely.
(ii) when $q \geq 2$, by Definition \ref{D-2.7}, there exists an
increasing sequence $\{r_m\}$$(r_m\to \infty)$
satisfying $(1+\frac{1}{m})r_m<r_{m+1}$, and
\[
\lim_{m\to \infty}\frac{\log_{p-1}T(r_m,f)}{(\log
_{q-1}r_m)^{\sigma_{(p.q)}(f)}}=\tau_{(p.q)}(f).
\]
Then there exists a positive constant $m_0$ such that for all
$m>m_0$ and for any given $\epsilon$ $
(0<\epsilon<\tau_{(p.q)}(f)-\beta)$ we have
\begin{equation} \label{E-3}
\log_{p-1}T(r_m,f)>(\tau_{(p.q)}(f)-\epsilon)(\log
_{q-1}r_m)^{\sigma_{(p.q)}(f)}.
\end{equation}
For any $r \in [r_m,(1+\frac{1}{m})r_m]$, we have
\[
\lim_{r_m\to +\infty}\frac{\log _{q-1}r_m}{\log _{q-1}r}=1.
\]
Since $\beta<\tau_{(p.q)}(f)-\epsilon$, there exists a positive
constant $m_1$ such that for all $m>m_1$, we have
\[
(\frac{\log _{q-1}r_m}{\log
_{q-1}r})^{\sigma_{(p.q)}(f)}>\frac{\beta}{\tau_{(p.q)}(f)-\epsilon};
\]
i.e.,
\begin{equation}\label{E-4}
(\tau_{(p.q)}(f)-\epsilon){(\log
_{q-1}r_m)}^{\sigma_{(p.q)}(f)}>\beta{(\log
_{q-1}r)}^{\sigma_{(p.q)}(f)}.
\end{equation}
Now we take $m_2=\max\{m_0,m_1\}$ and
$E_4=\cup_{m=m_2}^{\infty}[r_m,(1+\frac{1}{m})r_m]$, then by
\eqref{E-3}-\eqref{E-4}, for any $r \in E_4$, we have
\begin{align*}
\log_{p-1}T(r, f) 
&\geq \log_{p-1}T(r_m,f)\\
&>(\tau_{(p.q)}(f)-\epsilon){(\log _{q-1}r_m)}^{\sigma_{(p.q)}(f)}\\
&>\beta{(\log _{q-1}r)}^{\sigma_{(p.q)}(f)},
\end{align*}
where
\[
m_{l}E_4=\Sigma_{m=m_2}^{\infty}\int_{r_m}^{(1+\frac{1}{m})r_m}\frac{dt}{t}
=\Sigma_{m=m_2}^{\infty}\log (1+\frac{1}{m})=\infty. \qedhere
\]
\end{proof}

\begin{lemma}[\cite{20}]\label{L-9}
 Let $g(r)$  and $h(r)$ be monotone nondecreasing functions on $[0,\infty)$
such that $g(r) \leq h(r)$ for all $r \not\in[0,1]\cup E_{5}$, where
 $E_{5}\in (1,\infty)$ is a set
of finite logarithmic measure. Then for any constant 
$\alpha >1$, there exists $r_0=r_0(\alpha)>0$ such that
$g(r) \leq h(\alpha r)$ for all $r\geq r_0$.
\end{lemma}

\begin{lemma}\label{L-10}
Let $A_0,A_1,\dots,A_{k-1}$, $F\not\equiv 0$ be meromorphic functions
and let $f$ be a meromorphic solution of equation \eqref{E-2}. If
\[
\max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(F):j=0,1,\dots,k-1\}
<\sigma_{(p+1,q)}(f),
\]
then we have
\[
\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f).
\]
\end{lemma}

\begin{proof}
By \eqref{E-1}, we have
\begin{equation}\label{E-5}
\frac{1}{f}=\frac{1}{F}(\frac{f^{(k)}}{f}+A_{k-1}\frac{f^{(k-1)}}{f}+\dots +A_0).
\end{equation}
It is easy to see that if $f$ has a zero at $z_0$ of order $\beta$
$(\beta >k)$ and if $A_0,A_1,\dots,A_{k-1}$ are all analytic at
$z_0$, then $F$ has a zero at $z_0$ of order at least $\beta-k$. Hence
\begin{gather}\label{E-6}
n(r ,\frac{1}{f}) \leq k \overline n(r ,\frac{1}{f})+n(r
,\frac{1}{F})+\Sigma^{k-1}_{j=0}n(r ,A_j),
\\  \label{E-7}
N(r ,\frac{1}{f}) \leq k \overline N(r ,\frac{1}{f})+N(r
,\frac{1}{F})+\Sigma^{k-1}_{j=0}N(r ,A_j).
\end{gather}
By the lemma of the logarithmic derivative and \eqref{E-5}, we have 
\begin{equation}\label{E-8}
m(r ,\frac{1}{f}) \leq m(r ,\frac{1}{F})+\Sigma^{k-1}_{j=0}m(r
,A_j)+O(\log T(r,f)+ \log r)
\end{equation}
holds for all $|z|=r\not\in E_{6}$, where $E_{6}$ is a set of finite
linear measure. By \eqref{E-7},\eqref{E-8} and the first main theorem, we have
\begin{equation}\label{E-9}
T(r,f)=T(r,\frac{1}{f})+O(1)\leq k \overline N(r
,\frac{1}{f})+T(r,F)+\Sigma^{k-1}_{j=0}T(r ,A_j)+O(\log(r
T(r,f)))
\end{equation}
holds for all sufficiently $r \not\in E_{6}$.

Assume that
$\max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(F):j=0,1,\dots,k-1\}
<\sigma_{(p+1,q)}(f)$.
By Lemma \ref{L-6}, there exists a sequence $\{r_n\}$, $r_n\not\in E_{6}$ such that
\[
\lim_{r_n\to \infty}\frac{\log_{p+1}T(r_n,f)}{\log_qr_n
}=\sigma_{(p+1,q)}(f)=:\sigma_1.
\]
Hence , if $r_n \not\in E_{6}$ is sufficiently large, since
$\sigma_1>0$, then we have
\begin{equation}\label{E-10}
T(r_n,f) \geq \exp_{p+1}\{(\sigma_1-\epsilon) \log_qr_n\}
\end{equation}
holds for any given $\epsilon$ $(0 <2\epsilon <\sigma_1-\sigma_2)$, where
$\sigma_2=\max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(F):j=0,1,\dots,k-1\}$.
We have
\begin{equation}\label{E-11}
\max\{T(r_n,F),T(r_n,A_j):j=0,1,\dots,k-1\} \leq
\exp_{p+1}\{(\sigma_2+\epsilon) \log_qr_n\}.
\end{equation}
Since $\epsilon$ $(0 <2\epsilon <\sigma_1-\sigma_2)$, then from 
\eqref{E-10} and \eqref{E-11} we obtain 
\begin{equation}\label{E-12}
\max\{\frac{T(r_n,F)}{T(r_n,f)},\frac{T(r_n,A_j)}{T(r_n,f)}:
j=0,1,\dots,k-1\}\to0 \quad (r_n \to \infty).
\end{equation}
For sufficiently large $r_n$, we have
$$
O(\log (r_nT(r_n, f))) = O (T (r_n, f)).
$$
Hence, by \eqref{E-9} and \eqref{E-12}, we obtain that for sufficiently 
large $r_n\not\in E_{6}$, there holds
\[
(1-o(1))T(r_n,f) \leq k \overline {N}(r_n,\frac{1}{f}).
\]
Then we have $\overline{\lambda}_{(p+1,q)}(f) \geq
\sigma_{(p+1,q)}(f)$, and by definitions we have
$\overline{\lambda}_{(p+1,q)}(f)\leq \lambda_{(p+1,q)}(f)\leq
\sigma_{(p+1,q)}(f)$. Therefore
\[
\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f). \qedhere
\]
\end{proof}

\section{Proofs of Theorems \ref{T-1}-\ref{T-6}}

\subsection*{Proof of Theorem \ref{T-1}}

We shall divide the proof into two parts.

$\bullet$ Firstly, we prove that $\sigma_{(p+1,q)}(f)\leq
\sigma_{(p,q)}(A_s)\leq \sigma_{(p,q)}(f)$ holds for every
transcendental meromorphic function $f$ of \eqref{E-1}.
 By \eqref{E-1}, we know
that the poles of $f$ can only occur at the poles of $A_0, A_1,
\dots, A_{k-1}$, note that the multiplicities of poles of $f$ are
uniformly bounded, then we have
\begin{align*}
N(r,f) 
&\leq  C_1\overline{N}(r,f) \\
&\leq C_1\Sigma_{j=0}^{k-1}\overline{N}(r,A_j)\\
&\leq C_2\max\{N(r,A_j): j=0,1,\dots,k-1\} \leq O(T(r,A_s)),
\end{align*}
where $C_1$ and $C_2$ are suitable positive constants. Then we
have
\begin{equation}\label{E-13}
\log T(r,f)\leq\log m(r,f)+\log N(r,f)+\log 2\leq \log m(r,f)+ O\{\log T(r,A_s)\}.
\end{equation}
By \eqref{E-13} and Lemma \ref{L-2}, we obtain 
\begin{align*}
\log T(r,f)
&\leq  \log m(r,f)+ O\{\log T(r,A_s)\}\\
&= O\left(T(r, A_s)\{(\log r)\log T(r, A_s)\}^{\lambda}\right)
\end{align*}
outside of an exceptional set $E_0$ with $\int_{E_0}\frac{dt}{t}<\infty$,
this implies
$\sigma_{(p+1,q)}(f)\leq \sigma_{(p,q)}(A_s)$.
On the other hand, by \eqref{E-1}, we obtain
\begin{align*}
-A_s
&=\frac{f^{(k)}}{f^{(s)}}+A_{k-1}\frac{f^{(k-1)}}{f^{(s)}}+\dots
 +A_{s+1}\frac{f^{(s+1)}}{f^{(s)}}+
A_{s-1}\frac{f^{(s-1)}}{f^{(s)}}+\dots+A_0\frac{f}{f^{(s)}}
\\
&=\frac{f}{f^{(s)}}\{\frac{f^{(k)}}{f}+A_{k-1}\frac{f^{(k-1)}}{f}+\dots
 +A_{s+1}\frac{f^{(s+1)}}{f}+
A_{s-1}\frac{f^{(s-1)}}{f}+\dots+A_0\}.
\end{align*}
Since 
\[
m(r,\frac{f}{f^{(s)}})\leq
T(r,f)+T(r,\frac{1}{f^{(s)}})=T(r,f)+T(r,f^{(s)})+O(1)=O(T(r,f)),
\]
then by the lemma of logarithmic derivative we have
\begin{equation}\label{E-14}
T(r ,A_s)\leq N(r ,A_s)+\Sigma_{j\neq s}m(r ,A_j)+O(\log r
T(r,f))+O(T(r,f))
\end{equation}
hold for all $|z|=r\not\in E_{7}$, where $E_{7}$ is a set of finite
linear measure. By Lemma \ref{L-6} and similar discussion as in the
proof of Lemma \ref{L-10}, we see that there exists a sequence
$\{r_n\}$ $(r_n\to \infty)$ such that
\[
\sigma_1:=\sigma_{(p,q)}(A_s)=\lim_{r_n\to
\infty}\frac{\log_pT(r_n, A_s)}{\log_qr_n }
\]
and
\begin{gather}\label{E-15}
T(r_n, A_s) \geq \exp_p\{(\sigma_1-\epsilon) \log_qr_n\},\\
\label{E-16}
N(r_n,A_s) \leq \exp_p\{(\sigma_2+\epsilon) \log_qr_n\}, \\
\label{E-17}
 m(r_n, A_j)\leq \exp_p\{(\sigma_2+\epsilon) \log_qr_n\}\quad (j\neq s),
\end{gather}
where $\sigma_2:=\max\{\sigma_{(p,q)}(A_j),
\lambda_{(p,q)}(\frac{1}{A_s}): j\neq s\}$ and
 $0<2\epsilon<\sigma_1-\sigma_2$.

By \eqref{E-14}-\eqref{E-17}, we obtain
\[
(1-o(1))\exp_p\{(\sigma_1-\epsilon) \log_qr_n\}\leq
O\{\log r_n T(r_n,f)\}+O(T(r_n,f)).
\]
Hence we have $\sigma_{(p,q)}(A_s)=\sigma_1\leq \sigma_{(p,q)}(f)$.

$\bullet$ Secondly, we prove that  there exists at least one meromorphic
solution that satisfies
 $$
\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_s).
$$
Now we can assume that $\{f_1,f_2,\dots,f_{k} \}$ is a
meromorphic solution base of \eqref{E-1}. By Lemma \ref{L-1},
\[
m(r,A_s) \leq O\left(\log(\max_{1\leq n\leq k}T(r,f_n))\right).
\]
Now we assert that $m(r,A_s)>N(r,A_s)$ holds for sufficiently
large $r$. Indeed, if $m(r,A_s) \leq N(r,A_s)$, then
\[
T(r,A_s)=m(r,A_s)+N(r,A_s) \leq 2 N(r,A_s),
\]
so
\[
\limsup_{r\to \infty}\frac{\log_p T(r,A_s)}{\log_q r } \leq
\limsup_{r\to \infty}\frac{\log_p 2 N(r,A_s)}{\log_q r },
\]
then we have $\sigma_{(p,q)}(A_s) \leq \lambda_{(p,q)}(\frac{1}{A_s})$,
which contradicts the condition 
$\lambda_{(p,q)}(\frac{1}{A_s})<\sigma_{(p,q)}(A_s)$. Hence, 
$$
T(r,A_s)=O(m(r,A_s))\leq O\Big(\log(\max_{1\leq n\leq k}T(r,f_n))\Big).
$$
By Lemma \ref{L-6}, there exists a set $E_9\subset (0,\infty)$ has
finite linear measure , and a sequence $\{r_n\}$, $r_n\not\in
E_9$, such that
\[
\lim_{r_n\to \infty}\frac{\log_pT(r_n,A_s)}{\log_qr_n
}=\sigma_{(p,q)}(A_s).
\]
Set
\[
 T_n=\{r:r \in (0,\infty)\setminus E_9,\quad T(r,A_s) \leq
O(\log(T(r,f_n)))\quad (n=1,2,\dots,k)
\]
By Lemma \ref{L-1}, we have $\cup_{n=1}^{k}T_n=(0,\infty)\setminus E_9$.
It is easy to see that there exists at least one $T_n$, say
$T_1\subset (0,\infty)\setminus E_9$, that has infinite linear
measure and satisfies
\begin{equation}\label{E-18}
T(r,A_s) \leq O(\log T(r,f_1)).
\end{equation}
From \eqref{E-18}, we have $\sigma_{(p+1,q)}(f_1) \geq \sigma_{(p,q)}(A_s)$.

In the first part we have proved that $\sigma_{(p+1,q)}(f_1) \leq
\sigma_{(p,q)}(A_s)$. Therefore, we have that there is at least one
meromorphic solution $f_1$ satisfies
\[
\sigma_{(p+1,q)}(f_1)=\sigma_{(p,q)}(A_s).
\]

\subsection*{Proof of Theorem \ref{T-2}}

Suppose that $f$ is a nonzero meromorphic solution whose poles are of
uniformly bounded multiplicities of \eqref{E-1}, then \eqref{E-1}
can be written
\begin{equation}\label{E-19}
-A_0=\frac{f^{(k)}}{f}+ A_{k-1}\frac{f^{(k-1)}}{f}+\dots
+A_1\frac{f'}{f}.
\end{equation}
By the lemma of the logarithmic derivative and \eqref{E-19}, we have
\begin{equation}\label{E-20}
\begin{split}
 m(r,A_0)
&\leq \sum_{j=1}^{k-1}m(r,A_j)+\sum_{j=1}^{k}m(r,\frac{f^{(j)}}{f})+O(1)\\
&=\sum_{j=1}^{k-1}m(r,A_j)+O\{\log (r T(r,f))\}
\end{split}
\end{equation}
holds for all sufficiently large $r \not\in E_{10}$, where
$E_{10}\subset (0,\infty)$  has finite linear measure. Hence
\begin{equation}\label{E-21}
T(r,A_0)=m(r,A_0)+N(r,A_0) \leq
N(r,A_0)+\sum_{j=1}^{k-1}m(r,A_j)+O\{\log (r T(r,f))\}
\end{equation}
holds for all sufficiently large $|r|=r \not\in E_{10}$.

Since $\max\{\sigma_{(p,q)}(A_j):j\neq 0\}<\sigma_{(p,q)}(A_0)<\infty$,
by Lemma \ref{L-5}, there exist a set $E_{11}\subset (1,\infty)$
having infinite logarithmic measure such that for all $z$ satisfying
$|z|=r \in E_{11}$, we have
\begin{equation}\label{E-22}
\lim_{r\to \infty}\frac{\log_pT(r, A_0)}{\log_qr
}=\sigma_{(p,q)}(A_0),\, \frac{m(r,A_j)}{m(r,A_0)}=o(1)\quad(r \in E_{10},
j=1,2,\dots,k-1).
\end{equation}
By \eqref{E-20} and \eqref{E-22}, for all sufficiently large 
$r \in E_{11}\setminus E_{10}$, we have
\begin{equation}\label{E-23}
\frac{1}{2} m(r,A_0)\leq O\{\log (r T(r,f))\}.
\end{equation}
Using a similar discussion as in second part of proof of Theorem
\ref{T-1}, we can get that
\begin{equation}\label{E-24}
m(r,A_0)>N(r,A_0),
\end{equation}
hence, 
$$
T(r, A_0)=m(r,A_0)+N(r, A_0)=O(m(r, A_0))=O(\log rT(r, f))
$$ 
for all sufficiently large $r \in E_{11}\setminus E_{10}$, this means
\[
\sigma_{(p+1,q)}(f) \geq \sigma_{(p,q)}(A_0).
\]
On the other hand, by Theorem \ref{T-1}, we have
\[
\sigma_{(p+1,q)}(f) \leq \sigma_{(p,q)}(A_0).
\]
Therefore, every meromorphic solution $f$ whose poles are of
uniformly bounded multiplicities of \eqref{E-1} satisfies
\[
\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0). 
\]


\subsection*{Proof of Theorem \ref{T-3}}
When $A_0,A_1,\dots,A_{k-1}$ satisfy
\[
\max\{\sigma_{(p,q)}(A_j):j\neq 0\}<\sigma_{(p,q)}(A_0),
\]
then by Theorem \ref{T-2}, it is easy to see that Theorem \ref{T-3} holds.
Now we assume that there exists at least one of $A_j$
$(j=1,2,\dots,k-1)$ satisfies
$\sigma_{(p,q)}(A_j)=\sigma_{(p,q)}(A_0)$.

Suppose that $f$ is a nonzero meromorphic solution of \eqref{E-1}, we have
\begin{equation}\label{E-25}
|A_0| \leq |\frac{f^{(k)}(z)}{f(z)}|+
|A_{k-1}||\frac{f^{(k-1)}(z)}{f(z)}|+ \dots
+|A_1||\frac{f'(z)}{f(z)}|.
\end{equation}
Using a similar discussion as in the proof of Theorem \ref{T-2}, we
can get that \eqref{E-20} and \eqref{E-21} hold for all sufficiently large 
$r \not\in E_{12}$, where $E_{12}\subset (0,\infty)$  has finite linear
measure.
Since
\[
\max\{\sigma_{(p,q)}(A_j):j=1,2,\dots,k-1\}=\sigma_{(p,q)}(A_0)
\]
and
\[\max\{\tau_{(p,q)}(A_j):\sigma_{(p,q)}(A_j)
=\sigma_{(p,q)}(A_0)>0\}<\tau_{(p,q)}(A_0),
\]
then there exists a set $J \subset\{1,2,\dots,k-1\}$ such that for
$j \in J$, we have $\sigma_{(p,q)}(A_j)=\sigma_{(p,q)}(A_0)$ and
$\tau_{(p,q)}(A_j)<\tau_{(p,q)}(A_0)$.

Hence, there exist two  constants $\beta_1$ and $\beta_2$
satisfying $\max\{\tau_{(p,q)}:j \in
J\}<\beta_1<\beta_2\leq\tau_{(p,q)}(A_0)$. By Definitions
\ref{D-2.6} and \ref{D-2.7}, we obtain that
\begin{equation}\label{E-26}
m(r,A_j) \leq T(r,A_j)<\exp_{p-1}\{\beta_1
(\log_{q-1}r)^{\sigma_{(p,q)}(A_0)}\}.
\end{equation}
Since $\lambda_{(p,q)}(\frac{1}{A_0})<\sigma_{(p,q)}(A_0)$, we
have
\begin{equation}\label{E-27}
N(r,A_0) \leq
\exp_p\{(\lambda_(\frac{1}{A_0})+\epsilon)\log_qr \} \leq
\exp_{p-1}\{\beta_1 (\log_{q-1}r)^{\sigma_{(p,q)}(A_0)}\}.
\end{equation}
By Lemma \ref{L-8}, there exists a set of $E_{13}$ having infinite
logarithmic measure such that for all $r\in E_{13}$, we have
\begin{equation}\label{E-28}
T(r,A_0) \geq \exp_{p-1}\{\beta_2
(\log_{q-1}r)^{\sigma_{(p,q)}(A_0)}\}.
\end{equation}

Now, substituting \eqref{E-26}-\eqref{E-28}into \eqref{E-21}, we have
 $$
(1-o(1))\exp_{p-1}\{\beta_2(\log_{q-1}r)^{\sigma_{(p,q)(A_0)}}\}
\leq O(\log(r T(r, f)))
$$ 
for all $r\in E_{13}\setminus E_{12}$,
this implies 
\[
\sigma_{(p+1,q)}(f) \geq \sigma_{(p,q)}(A_0).
\]
On the other hand ,by Theorem \ref{T-1}, we have
\[
\sigma_{(p+1,q)}(f) \leq \sigma_{(p,q)}(A_0).
\]
Then we have that
\[
\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0)
\]
holds for any nonzero meromorphic solution $f$ whose poles are of
uniformly bounded multiplicities of \eqref{E-1}.

\subsection*{Proof of Theorem \ref{T-4}}

Since all solutions of equation \eqref{E-2} are meromorphic functions, all
solutions of the homogeneous differential equation \eqref{E-1} corresponding
to equation \eqref{E-2} are still  meromorphic functions.

Now we assume that $\{f_1,f_2,\dots,f_{k}\}$ is a meromorphic
solution base of \eqref{E-1}, then by the elementary theory of differential
equations (see, e.g. \cite{12}), any solution of \eqref{E-2} has the form
\begin{equation}\label{E-29}
f=c_1(z)f_1+c_2(z)f_2+ \dots +c_{k}(z)f_{k},
\end{equation}
where $c_1,c_2,\dots,c_{k}$ are suitable meromorphic functions
satisfying
\begin{equation}\label{E-30}
c'_j=FG_j(f_1,f_1,\dots,f_{k})
W(f_1,f_1,\dots,f_{k})^{-1}\quad (j=1,2,\dots,k),
\end{equation}
where $G_j(f_1,f_1,\dots,f_{k})$ are differential polynomials
in $\{f_1,f_2,\dots,f_{k}\}$ and their derivatives, and
$W(f_1,f_1,\dots,f_{k})^{-1}$ is the Wronskian of
$\{f_1,f_2,\dots,f_{k}\}$.
By Theorem \ref{T-2}, we have
\[
\sigma_{(p+1,q)}(f_j)=\sigma_{(p,q)}(A_0)\quad (j=1,2,\dots,k).
\]
By Lemma \ref{L-3}, \eqref{E-29} and \eqref{E-30}, we obtain
\[
\sigma_{(p+1,q)}(f)\leq
\max\{\sigma_{(p+1,q)}(f_j),\sigma_{(p+1,q)}(F):j=1,2,\dots,k\}
= \sigma_{(p,q)}(A_0).
\]
Now we assert that all solutions $f$ of \eqref{E-2} satisfy
$\sigma_{(p+1,q)}(f)=\sigma_{(p,q)}(A_0)$ with at most one
exceptional solution, say $f_0$, satisfying
$\sigma_{(p+1,q)}(f_0)<\sigma_{(p,q)}(A_0)$. In fact, if there
exists two distinct meromorphic functions $f_0$ and $f_1$ of \eqref{E-2}
satisfying
\[
\sigma_{(p+1,q)}(f_j)<\sigma_{(p,q)}(A_0)\quad (j=0,1),
\]
then $f=f_0-f_1$ is a nonzero meromorphic solution of \eqref{E-1}, and
satisfying $\sigma_{(p+1,q)}(f)<\sigma_{(p,q)}(A_0)$, this
contradicts Theorem \ref{T-2}.

For all the solutions $f$ of \eqref{E-2} satisfying 
$\sigma_{(p+1,q)}(f)=\sigma_{(p, q)}(A_0)$, we have 
\[
\max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(F):j=0,1,\dots,k-1\}
<\sigma_{(p+1,q)}(f).
\] 
Thus by Lemma \ref{L-10}, we obtain
\[
\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f).
\]
Therefore, Theorem \ref{T-4} is proved.

\subsection{Proof of Theorem \ref{T-5}}
Suppose that $\{g_1,g_2,\dots,g_{k}\}$ is a meromorphic
solution base of \eqref{E-1} corresponding to \eqref{E-2}. By a similar discussion
as in the proof of Theorem \ref{T-4}, we obtain
\[
\sigma_{(p+1,q)}(f)\leq
\max\{\sigma_{(p+1,q)}(g_j),\sigma_{(p+1,q)}(F):j=1,2,\dots,k\}
\]
By the first part of the proof of Theorem \ref{T-1}, we can get that
\[
\sigma_{(p+1,q)}(g_j) \leq
\max\{\sigma_{(p,q)}(A_j):j=0,1,\dots,k-1\} \leq
\sigma_{(p+1,q)}(F),
\]
then we can get
\begin{equation}\label{E-31}
\sigma_{(p+1,q)}(f)\leq \sigma_{(p+1,q)}(F).
\end{equation}
On the other hand, by the simple order comparison from \eqref{E-2}, we have
\[
\sigma_{(p+1,q)}(F)\leq
\max\{\sigma_{(p+1,q)}(A_j),\sigma_{(p+1,q)}(f):j=0,1,\dots,k-1\}.
\]
Since $\sigma_{(p+1,q)}(A_j)< \sigma_{(p+1,q)}(F)$, we have
\begin{equation}\label{E-32}
\sigma_{(p+1,q)}(F)\leq \sigma_{(p+1,q)}(f).
\end{equation}
By \eqref{E-31}-\eqref{E-32}, we obtain 
\[
\sigma_{(p+1,q)}(F)=\sigma_{(p+1,q)}(f).
\]
Therefore, the proof of Theorem \ref{T-5} is complete.

\subsection*{Proof of Theorem \ref{T-6}}
(i) By the simple order comparison from \eqref{E-2} it is easy to see that all
meromorphic solutions of \eqref{E-2} satisfy
\[
\sigma_{(p+1,q)}(f)\geq \sigma_{(p+1,q)}(F).
\]
On the other hand, by the similar proof in \eqref{E-29}-\eqref{E-30}, we obtain that
all meromorphic solutions of \eqref{E-2} satisfy
\[
\sigma_{(p+1,q)}(f)\leq \sigma_{(p+1,q)}(F)
\]
if $\sigma_{(p+1,q)}(F)\geq \alpha_1$.
Therefore, all meromorphic solutions whose poles are of
uniformly bounded multiplicities  of \eqref{E-2} satisfy
\[
\sigma_{(p+1,q)}(f)= \sigma_{(p+1,q)}(F).
\]

(ii) By the hypotheses that
\[
|A_0{(z)}|\geq\exp_{p+1}\{(\alpha_1-\epsilon)\log_qr\},
\]
and $ |A_j{(z)}|\leq\exp_{p+1}\{\alpha_2\log_qr\}$, we can easily
obtain that $\sigma_{(p+1,q)}(A_0)= \alpha_1$. Since
$\sigma_{(p+1,q)}(F)< \alpha_1=\sigma_{(p+1,q)}(A_0)$, by the
similar proof in Theorem \ref{T-4}, we obtain that all meromorphic
solutions whose poles are of
uniformly bounded multiplicities  of \eqref{E-2} satisfy
\[
\overline{\lambda}_{(p+1,q)}(f)=\lambda_{(p+1,q)}(f)=\sigma_{(p+1,q)}(f)=\alpha_1
\]
with at most one exceptional solution $f_2$ satisfying
$\sigma_{(p+1,q)}(f_2)<\alpha_1$. Therefore, we completely prove 
Theorem \ref{T-6}.

\subsection*{Acknowledgements} 
This research was supported by grants 11101201 from the NSFC, and 2010GQS0139 
from the NSF  of Jiangxi of China.
The authors would like to thank the anonymous referee for making valuable 
suggestions and comments to improve the present paper.


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\end{document}