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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 204, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/204\hfil Growth of solutions]
{Growth of solutions of linear differential equations with
analytic coefficients of $[p,q]$-order in the unit disc}

\author[H. Hu, X.-M. Zheng \hfil EJDE-2014/204\hfilneg]
{Hui Hu, Xiu-Min Zheng}  % in alphabetical order

\address{Hui Hu \newline
Institute of Mathematics and Information Science,
Jiangxi Normal University, 330022, China}
\email{h\_h87\_6@hotmail.com}

\address{Xiu-Min Zheng (corresponding author) \newline
Institute of Mathematics and Information Science,
Jiangxi Normal University, 330022, China}
\email{zhengxiumin2008@sina.com}

\thanks{Submitted October 23, 2013. Published September 30, 2014.}
\subjclass[2000]{30D35, 34M10}
\keywords{Unit disc; analytic function;
(lower) $[p,q]$-order; (lower)  $[p,q]$-type; \hfill\break\indent 
(lower) $[p,q]$-convergence exponent}

\begin{abstract}
 In this article, we study the growth of solutions of homogeneous
 linear differential equation in which the coefficients are analytic
 functions of $[p,q]$-order in the unit disc.
 We obtain results about the (lower) $[p,q]$-order of the
 solutions, and the (lower) $[p,q]$-convergence exponent for the 
 sequence of distinct zeros of $f(z)-\varphi(z)$.
\end{abstract}

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

\section{Introduction}

We study the growth of solutions of the following two equations, 
for $n\geq 2$,
\begin{equation}
f^{(n)}+A_{n-1}(z)f^{(n-1)}+\dots+A_1(z)f'+A_0(z)f=0, \label{e1.1}
\end{equation}
and
\begin{equation}
f^{(n)}+A_{n-1}(z)f^{(n-1)}+\dots+A_1(z)f'+A_0(z)f=F(z), \label{e1.2}
\end{equation}
where $A_n(z), \dots, A_1(z), A_0(z)$ $(\not\equiv 0)$ and
$F(z)(\not\equiv 0)$ are meromorphic functions in the complex plane
$\mathbb{C}$ or in the unit disc $\Delta=\{z\in\mathbb{C}: |z|<1\}$.

As is well known, the theory of meromorphic solutions of linear
differential equations \eqref{e1.1} and \eqref{e1.2} in $\mathbb{C}$ becomes
mature and the results are fruitful, but the theory of meromorphic
solutions of equations \eqref{e1.1} and \eqref{e1.2} in $\Delta$ is not as
developed as the one in $\mathbb{C}$. The reason may be that the
properties of meromorphic functions in $\mathbb{C}$ or $\Delta$ are
of some differences, which result in that some important tools in
$\mathbb{C}$ are ineffective in $\Delta$. However, we also discover
that for meromorphic solutions of equations \eqref{e1.1} and \eqref{e1.2} in
$\Delta$, there are many similar properties as the ones of
meromorphic solutions of equations \eqref{e1.1} and \eqref{e1.2} in $\mathbb{C}$.
For example, it is well known that when the coefficients are entire
functions, the solutions of equations \eqref{e1.1} and \eqref{e1.2} are entire
functions. Similarly, when the coefficients are analytic functions
in $\Delta$, the solutions of equations \eqref{e1.1} and \eqref{e1.2} are analytic
functions in $\Delta$, and there are exactly $n$ linearly
independent solutions of equation \eqref{e1.1} (see e.g. \cite{h2}). Hence,
it is interesting to investigate meromorphic solutions of linear
differential equations \eqref{e1.1} and \eqref{e1.2} in $\Delta$.

Typically,  Heittokangas \cite{h2} investigated meromorphic
solutions of linear differential equations \eqref{e1.1} and \eqref{e1.2} in
$\Delta$ by introducing the definition of the function spaces and
his results also gave some important tools for further
investigations on the theory of meromorphic solutions of equations
\eqref{e1.1} and \eqref{e1.2} in $\Delta$. After that, many papers (see e.g.
\cite{b2, b3, c1, c2, h3, s1}) focused on this topic. We proceed in
this way in this paper, inspired by the relative case in
$\mathbb{C}$; that is, we try to find some results similar to the
one in Liu-Tu-Shi \cite{l3}, which is stated as follows.


\begin{theorem}[\cite{l3}] \label{thmA}
Let $A_0(z), \dots,A_{n-1}(z)$ be entire functions satisfying
\begin{gather*}
\max\{\sigma_{[p,q]}(A_j):j=1,\dots, n-1\}\leq\sigma_{[p,q]}(A_0)<\infty,\\
\max\{\tau_{[p,q]}(A_j):\sigma_{[p,q]}(A_j)=\sigma_{[p,q]}(A_0)>0,j\neq
0\}<\tau_{[p,q]}(A_0).
\end{gather*}
Then every nontrivial solution $f(z)$ of
\eqref{e1.1} satisfies $\sigma_{[p+1,q]}(f)=\sigma_{[p,q]}(A_0)$.
\end{theorem}

Liu-Tu-Shi \cite{l3} used the $[p,q]$-type of $A_0(z)$ to dominate
the $[p,q]$-types of other coefficients, and got the result about
$\sigma_{[p+1,q]}(f)$. Thus, the following questions arise
naturally:
(1) Whether the results similar to Theorem \ref{thmA} can be
obtained in $\Delta$?
(2) If we use the lower $[p,q]$-type of
$A_0(z)$ to dominate other coefficients, what can be said about
$\mu_{[p+1,q]}(f)$?
(3) Can we find some other conditions to
dominate other coefficients? In this paper, we give some answers to
the above questions.

Before we give our main results in the next section, it is necessary
to  introduce some notation. In this paper, we
assume that the readers are familiar with the standard notation and
the fundamental results of the Nevanlinna's theory in $\mathbb{C}$
and $\Delta$ (see e.g. \cite{h1,h2,l1,l2}). Moreover, in
\cite{j1,j2},  Juneja and his co-authors investigated some
properties of entire functions of $[p,q]$-order, and obtained some
results. In \cite{l3}, in order to keep accordance with the general
definition of an entire function $f(z)$ of iterated $p$-order,
Liu-Tu-Shi gave a minor modification to the original definition of
$[p,q]$-order given in \cite{j1,j2}. Further, in \cite{b2,b3},
Bela\"{\i}di defined  $[p,q]$-order of analytic and
meromorphic functions in $\Delta$. For conveniences, we list the
following concepts (see e.g. \cite{b2, b3, h4}).

\begin{definition} \label{def1.1} \rm
Let $p,q$ be integers such that $p\geq q\geq1$, and $f(z)$ be a meromorphic
function in $\Delta$.
The $[p,q]$-order and the lower $[p,q]$-order of $f(z)$ are defined
respectively by
$$
\sigma_{[p,q]}(f)=\limsup_{r\to 1^-}
\frac{\log_p^+ T(r,f)}{\log_q\frac{1}{1-r}},\quad
\mu_{[p,q]}(f)=\liminf_{r\to 1^-} 
\frac{\log_p^+ T(r,f)}{\log_q\frac{1}{1-r}}.
$$
For an analytic function $f(z)$ in $\Delta$, we also define
$$
\sigma_{M,[p,q]}(f)=\limsup_{r\to 1^-}
\frac{\log_{p+1}^+ M(r,f)}{\log_q\frac{1}{1-r}},\quad
\mu_{M,[p,q]}(f)=\liminf_{r\to 1^-} 
\frac{\log_{p+1}^+ M(r,f)}{\log_q\frac{1}{1-r}}.
$$
\end{definition}

\begin{definition} \label{def1.2}\rm
Let $p,q$ be integers such that
$p\geq q\geq1$, $a\in\mathbb{C}\bigcup\{\infty\}$, and $f(z)$ be a
meromorphic function in $\Delta$. The $[p,q]$-convergence exponents
of the sequence of $a$-points and the sequence of distinct
$a$-points of $f(z)$ are defined respectively by
$$
\lambda_{[p,q]}(f-a)=\limsup_{r\to 1^-}
\frac{\log_p^+  N(r,\frac{1}{f-a})}{\log_q\frac{1}{1-r}},\quad
\overline{\lambda}_{[p,q]}(f-a)=\limsup_{r\to 1^-} \frac{\log_p^+
\overline{N}(r,\frac{1}{f-a})}{\log_q\frac{1}{1-r}}.
$$
The lower $[p,q]$-convergence exponents of the sequence of $a$-points and the
sequence of distinct $a$-points of $f(z)$ are defined respectively by
\begin{gather*}
\underline{\lambda}_{[p,q]}(f-a)=\liminf_{r\to 1^-}
 \frac{\log_p^+ N(r,\frac{1}{f-a})}{\log_q\frac{1}{1-r}}, \\
\overline{\underline{\lambda}}_{[p,q]}(f-a)
=\liminf_{r\to 1^-}\frac{\log_p^+
\overline{N}(r,\frac{1}{f-a})}{\log_q\frac{1}{1-r}}.
\end{gather*}
Furthermore, we obtain the definitions of
$\lambda_{[p,q]}(f-\varphi)$,
$\overline{\lambda}_{[p,q]}(f-\varphi)$,
$\underline{\lambda}_{[p,q]}(f-\varphi)$ and
$\overline{\underline{\lambda}}_{[p,q]}(f-\varphi)$ in $\Delta$,
when the constant $a$ in Definiton \ref{def1.2} is replaced by a meromorphic
function $\varphi(z)$ in $\Delta$.
\end{definition}

\begin{definition} \label{def1.3}\rm
Let $p,q$ be integers such that
$p\geq q\geq1$, and $f(z)$ be a meromorphic function of
$[p,q]$-order $\sigma (0<\sigma<\infty)$ and lower $[p,q]$-order
$\mu (0<\mu<\infty)$ in $\Delta$. The $[p,q]$-type and the lower
$[p,q]$-type of $f(z)$ are defined respectively by
$$
\tau_{[p,q]}(f)=\limsup_{r\to 1^-}
\frac{\log_{p-1}^+ T(r,f)}{(\log_{q-1} \frac{1}{1-r})^{\sigma}},\quad
\underline{\tau}_{[p,q]}(f)=\liminf_{r\to 1^-}
\frac{\log_{p-1}^+ T(r,f)}{(\log_{q-1} \frac{1}{1-r})^{\mu}}.
$$
For an analytic function $f(z)$ in $\Delta$, we also define
$$
\tau_{M,[p,q]}(f)=\limsup_{r\to 1^-} \frac{\log_p^+ M(r,f)}{(\log_{q-1}
\frac{1}{1-r})^{\sigma}},\quad
\underline{\tau}_{M,[p,q]}(f)=\liminf_{r\to 1^-}
\frac{\log_p^+ M(r,f)}{(\log_{q-1} \frac{1}{1-r})^{\mu}}.
$$
\end{definition}

Different from the case in $\mathbb{C}$, we have the following
results for the case in $\Delta$.

\begin{proposition}[\cite{b2}] \label{prop1.1}
Let $p,q$ be integers
such that $p\geq q\geq1$, and $f(z)$ be an analytic function of
$[p,q]$-order in $\Delta$. The following two statements hold:
\begin{itemize}
\item[(i)] If $p=q$, then 
$\sigma_{[p,q]}(f)\leq\sigma_{M,[p,q]}(f)\leq\sigma_{[p,q]}(f)+1$.

\item[(ii)] If $p>q$, then $\sigma_{[p,q]}(f)=\sigma_{M,[p,q]}(f)$.
\end{itemize}
\end{proposition}

Similarly, we can get the following proposition.

\begin{proposition} \label{prop1.2}
Let $p,q$ be integers such that $p\geq q\geq1$, and $f(z)$ be an analytic
function of lower $[p,q]$-order in $\Delta$. The following two statements hold:
\begin{itemize}
\item[(i)] If $p=q$, then $\mu_{[p,q]}(f)\leq\mu_{M,[p,q]}(f)\leq\mu_{[p,q]}(f)+1$;

\item[(ii)] If $p>q$, then $\mu_{[p,q]}(f)=\mu_{M,[p,q]}(f)$.
\end{itemize}
\end{proposition}

\section{Main results}

In this paper, we consider the case that the coefficients are
analytic functions in $\Delta$, and obtain two main results on the
growth of solutions of equation \eqref{e1.1}. Moreover, we get the results
about the $[p,q]$-convergence exponent and the lower
$[p,q]$-convergence exponent of the sequence of distinct zeros of
$f(z)-\varphi(z)$.

\begin{theorem} \label{thm2.1}
Let $p,q$ be integers such that $p> q\geq2$,
and $A_{n-1}(z)$, \dots, $A_1(z)$, $A_0(z)$ $(\not\equiv 0)$ be analytic
functions in $\Delta$ with
$0<\mu=\mu_{[p,q]}(A_0)\leq\sigma_{[p,q]}(A_0)<\infty$. Assume that
$\max\{\sigma_{[p,q]}(A_j)|j=1,\dots,n-1\}\leq\mu_{[p,q]}(A_0)$ and that \\
$\max\{\tau_{[p,q]}(A_j)|\sigma_{[p,q]}(A_j)=\mu_{[p,q]}(A_0),j\neq
0\}<\underline{\tau}_{[p,q]}(A_0)=\tau<\infty$.
If $f(z)$ $(\not\equiv 0)$ is a solution of \eqref{e1.1}, then we have
\begin{align*}
\overline{\underline{\lambda}}_{[p+1,q]}(f-\varphi)
&=\mu_{[p+1,q]}(f)=\mu_{[p,q]}(A_0)\\
&\leq\sigma_{[p,q]}(A_0)=\sigma_{[p+1,q]}(f)
=\overline{\lambda}_{[p+1,q]}(f-\varphi),
\end{align*}
where $\varphi(z)$ $(\not\equiv 0)$ is an analytic function in $\Delta$
with $\sigma_{[p+1,q]}(\varphi)<\mu_{[p,q]}(A_0)$.
\end{theorem}

\begin{theorem} \label{thm2.2}
Let $p,q$ be integers such that $p>q\geq1$,
and $A_{n-1}(z)$, \dots, $A_1(z)$, $A_0(z)$ $(\not\equiv 0)$ be analytic
functions in $\Delta$ with
$0<\mu=\mu_{[p,q]}(A_0)\leq\sigma_{[p,q]}(A_0)<\infty$. Assume that
$\max\{\sigma_{[p,q]}(A_j)|j=1,\dots,n-1\}\leq\mu_{[p,q]}(A_0)$ and that\\
$\limsup _{r\to1^-} {\sum_{j=1}^{n-1}m(r,A_j)}/{m(r,A_0)}<1$.
If $f(z)$ $(\not\equiv 0)$ is a solution of \eqref{e1.1}, then we have
$$
\overline{\underline{\lambda}}_{[p+1,q]}(f-\varphi)
=\mu_{[p+1,q]}(f)=\mu_{[p,q]}(A_0)\leq\sigma_{[p,q]}(A_0)
=\sigma_{[p+1,q]}(f)=\overline{\lambda}_{[p+1,q]}(f-\varphi),
$$
where $\varphi(z)$ $(\not\equiv 0)$ is an analytic function in $\Delta$
with $\sigma_{[p+1,q]}(\varphi)<\mu_{[p,q]}(A_0)$.
\end{theorem}

\begin{remark} \label{rmk2.1}\rm
In Theorems \ref{thm2.1} and \ref{thm2.2}, we just consider the
case $p>q$ to make sure the  Lemmas \ref{lem3.8} and \ref{lem3.11} hold.
Moreover, $q\geq2$ in Theorem \ref{thm2.1} is necessary for using
Lemma \ref{lem3.10}.
\end{remark}

\section{Preliminary lemmas}

\begin{lemma}[\cite{h3}] \label{lem3.1}
Let $A_j(z)$, $j=0,\dots,n-1$ be analytic functions in $D_{R}$
$(D_R=\{z\in\mathbb{C}\big{|}|z|<R\})$, where $0<R\leq\infty$, and
$f(z)$ be a solution of \eqref{e1.1} in $D_R$, $1\leq p<\infty$. Then for
all $0\leq r<R$,
$$
m_p(r,f)^p\leq C\Big(\sum_{j=0}^{n-1}\int_0^{2\pi}
\int_{0}^r|A_j(se^{i\theta})|^{\frac{p}{n-j}}dsd\theta+1\Big),
$$
where $C=C(n)>0$ is a constant depending on $p$, and on the initial
values of $f(z)$ at the point $z_{\theta}$, where
$A_j(z_{\theta})\neq 0$ for some $j=0,\dots,n-1$.
\end{lemma}


\begin{lemma}[\cite{h2,s1}] \label{lem3.2}
Let $f(z)$ be a meromorphic function in $\Delta$, and $k\in \mathbb{N}$. Then
$$
m(r,\frac{f^{(k)}}{f})=S(r,f),
$$
where $S(r,f)=O(\log^+ T(r,f)+\log\frac{1}{1-r})$, possibly outside
a set $E_1\subset[0,1)$ with $\int_{E_1}\frac{dr}{1-r}<\infty$. If
$f(z)$ is of finite order (namely, finite iterated 1-order), then
$$
m(r,\frac{f^{(k)}}{f})=O(\log\frac{1}{1-r}).
$$
\end{lemma}

\begin{lemma}[\cite{b2}] \label{lem3.3}
Let $p,q$ be integers such that $p\geq q\geq1$, $k\geq 1$ be an integer and
$f(z)$ be a meromorphic function in $\Delta$ such that
$\sigma_{[p,q]}(f)=\sigma<\infty$. Then
$$
m(r,\frac{f^{(k)}}{f})=O\Big(\exp_{p-1}\{(\sigma+\varepsilon)
\log_q\frac{1}{1-r}\}\Big)
$$
holds for any $\varepsilon>0$ and all $r\to1^-$ outside a
set $E_2\subset[0,1)$ with $\int_{E_2}\frac{dr}{1-r}<\infty$.
\end{lemma}

\begin{lemma}[\cite{b1}] \label{lem3.4}
Let $g:(0,1)\to \mathbb{R}$ and $h:(0,1)\to \mathbb{R}$ be monotone
increasing functions such that $g(r)\leq h(r)$ holds outside an
exceptional set $E_3\subset[0,1)$ with
$\int_{E_3}\frac{dr}{1-r}<\infty$. Then there exists a constant
$d\in (0,1)$ such that if $s(r)=1-d(1-r)$, then $g(r)\leq h(s(r))$
for all $r\in [0,1)$.
\end{lemma}

\begin{lemma} \label{lem3.5}
Let $p,q$ be integers such that $p\geq q\geq1$
and $A_{n-1}(z)$, $\dots$, $A_1(z)$, $A_0(z)$ $(\not\equiv 0)$,
$F(z) $ $(\not\equiv 0)$ be meromorphic functions in $\Delta$. If $f(z)$
is a meromorphic solution of \eqref{e1.2} satisfying
$$
\max\{\sigma_{[p,q]}(F),\sigma_{[p,q]}(A_j)|j=0,\dots,n-1\}
<\sigma_{[p,q]}(f)=\sigma<\infty,
$$
then  $\overline{\lambda}_{[p,q]}(f)=\lambda_{[p,q]}(f)=\sigma_{[p,q]}(f)$.
\end{lemma}

\begin{proof}
By \eqref{e1.2}, we have
\begin{equation}
\frac{1}{f}=\frac{1}{F}\left(\frac{f^{(n)}}{f}
+A_{n-1}(z)\frac{f^{(n-1)}}{f}+\dots+A_0(z)\right).\label{e3.1}
\end{equation}
If $f(z)$ has a zero at $z_0\in \Delta$ of order $\gamma(>n)$ and
$A_{n-1}(z), \dots, A_1(z), A_0(z)$ are all analytic at $z_0$, then
$F(z)$ has a zero at $z_0$ of order at least $\gamma-n$. Hence, we
have
\begin{equation}
N(r,\frac{1}{f})\leq n\overline{N}(r,\frac{1}{f})
+N(r,\frac{1}{F})+\sum_{j=0}^{n-1}N(r,A_j).\label{e3.2}
\end{equation}
By \eqref{e3.1}, we have
\begin{equation}
m(r,\frac{1}{f})\leq m(r,\frac{1}{F})+\sum_{j=0}^{n-1}m(r,A_j)
+\sum_{j=1}^{n}m(r,\frac{f^{(j)}}{f})+O(1).~\label{e3.3}
\end{equation}
Lemma \ref{lem3.3} gives
\begin{equation}
m(r,\frac{f^{(j)}}{f})=O\Big(\exp_{p-1}\{(\sigma+\varepsilon)
\log_q\frac{1}{1-r}\}\Big),\quad j=1,\dots,n,\label{e3.4}
\end{equation}
holds for any $\varepsilon>0$ and all $r\to1^-$ outside a
set $E_2\subset[0,1)$ with $\int_{E_2}\frac{dr}{1-r}<\infty$.
Therefore, by \eqref{e3.2}-\eqref{e3.4} and the first fundamental theorem,
\begin{equation}
\begin{aligned}
T(r,f)&=T(r,\frac{1}{f})+O(1)\leq n\overline{N}(r,\frac{1}{f})+T(r,F)
+\sum_{j=0}^{n-1}T(r,A_j)\\
&\quad +O\Big(\exp_{p-1}\{(\sigma+\varepsilon)\log_q\frac{1}{1-r}\}\Big) 
\end{aligned}\label{e3.5}
\end{equation}
holds for all $r\to1^-,r\not\in E_2$. Set
$\rho=\max\{\sigma_{[p,q]}(F),\sigma_{[p,q]}(A_j)|j=0,\dots,n-1\}$,
then for $r\to1^-$, we have
\begin{equation}
\sum_{j=0}^{n-1}T(r,A_j)+T(r,F)\leq(n+1)\exp_{p}\{(\rho+\varepsilon)
\log_q\frac{1}{1-r}\}.\label{e3.6}
\end{equation}
Thus, by \eqref{e3.5} and \eqref{e3.6}, for all $r\to1^-$, $r\not\in E_2$,
we have
\begin{equation}
\begin{aligned}
T(r,f)
&\leq n\overline{N}(r,\frac{1}{f})+(n+1)\exp_{p}\{(\rho+\varepsilon)
\log_q\frac{1}{1-r}\} \\
&\quad +O\Big(\exp_{p-1}\{(\sigma+\varepsilon)\log_q\frac{1}{1-r}\}\Big)\\
&\leq n\overline{N}(r,\frac{1}{f})+\exp_{p}\{(\rho+2\varepsilon)
\log_q\frac{1}{1-r}\}. \label{e3.7}
\end{aligned}
\end{equation}
Hence, by Lemma \ref{lem3.4} and \eqref{e3.7}, for all $r\to 1^-$, we have
\begin{equation}
T(r,f)\leq n\overline{N}(s(r),\frac{1}{f})+\exp_{p}\{(\rho+2\varepsilon)
\log_q\frac{1}{1-s(r)}\},\label{e3.8}
\end{equation}
where $s(r)=1-d(1-r)$, $d\in (0,1)$. If
$\overline{\lambda}_{[p,q]}(f)<\sigma_{[p,q]}(f)=\sigma$, then for
any
$\varepsilon~(0<3\varepsilon<\sigma-\max\{\overline{\lambda}_{[p,q]}(f),\rho\})$
and all $r\to 1^-$, we have
\begin{align*}
T(r,f)&\leq n\exp_{p}\{(\overline{\lambda}_{[p,q]}(f)+\varepsilon)
 \log_q\frac{1}{1-s(r)}\}+\exp_{p}\{(\rho+2\varepsilon)\log_q\frac{1}{1-s(r)}\}\\
&\leq (n+1)\exp_{p}\{(\sigma-\varepsilon)\log_q\frac{1}{1-s(r)}\},
\end{align*}
which results in a contradiction that
$\sigma=\sigma_{[p,q]}(f)<\sigma-\varepsilon$. Therefore, we have
$\overline{\lambda}_{[p,q]}(f)\geq\sigma_{[p,q]}(f)=\sigma$. Since
$\overline{\lambda}_{[p,q]}(f)\leq\lambda_{[p,q]}(f)\leq\sigma_{[p,q]}(f)$,
the result holds.
\end{proof}

\begin{lemma} \label{lem3.6}
Let $p,q$ be integers such that $p\geq q\geq1$
and $A_{n-1}(z)$, $\dots$, $A_1(z)$, $A_0(z)$ $(\not\equiv 0)$,
$F(z)$ $(\not\equiv 0)$ be meromorphic functions in $\Delta$. If $f(z)$
is a meromorphic solution of \eqref{e1.2} satisfying
$$
\max\{\sigma_{[p,q]}(F),\sigma_{[p,q]}(A_j)|j=0,\dots,n-1\}
<\mu_{[p,q]}(f)\leq\sigma_{[p,q]}(f)<\infty,
$$
then we have
$\underline{\overline{\lambda}}_{[p,q]}(f)
=\underline{\lambda}_{[p,q]}(f)=\mu_{[p,q]}(f)$.
\end{lemma}

\begin{proof}
Since
$\max\{\sigma_{[p,q]}(F),\sigma_{[p,q]}(A_j)|j=0,\dots,n-1\}<\mu_{[p,q]}(f)$,
we have that for $r\to1^-$,
\begin{equation}
T(r,F)=o(T(r,f)),\quad T(r,A_j)=o(T(r,f)),\quad j=0,\dots,n-1.\label{e3.9}
\end{equation}
By \eqref{e3.5} and \eqref{e3.9}, we have
\begin{equation}
(1-o(1))T(r,f)\leq n\overline{N}(r,\frac{1}{f})
+O\Big(\exp_{p-1}\{(\sigma_{[p,q]}(f)+\varepsilon)\log_q\frac{1}{1-r}\}\Big),
\label{e3.10}
\end{equation}
for any $\varepsilon>0$ and $r\to1^-$, $r\not\in E_2$, where
$E_2\subset[0,1)$ satisfies $\int_{E_2}\frac{dr}{1-r}<\infty$.
Hence, by Lemma \ref{lem3.4} and \eqref{e3.10}, for all $r\to 1^-$, we have
$$
(1-o(1))T(r,f)\leq n\overline{N}(s(r),\frac{1}{f})
+O\Big(\exp_{p-1}\{(\sigma_{[p,q]}(f)+\varepsilon)\log_q\frac{1}{1-s(r)}\}\Big),
$$
where $s(r)=1-d(1-r)$, $d\in (0,1)$. Hence, we have
$\overline{\underline{\lambda}}_{[p,q]}(f)\geq\mu_{[p,q]}(f)$. Since
$\overline{\underline{\lambda}}_{[p,q]}(f)
\leq\underline{\lambda}_{[p,q]}(f)\leq\mu_{[p,q]}(f)$,
the result holds.
\end{proof}

\begin{lemma} \label{lem3.7}
Let $p,q$ be integers such that $p\geq q\geq1$
and $f(z)$ be an analytic function in $\Delta$ with
$\mu_{[p,q]}(f)=\mu<\infty$. Then for any given $\varepsilon>0$,
there exists a set $E_4\subset[0,1)$ with
$\int_{E_4}\frac{dr}{1-r}=\infty$, such that
$$
\mu=\mu_{[p,q]}(f)=\lim_{ r\to1^-,\,r\in E_4}
\frac{\log_{p}^+ T(r,f)}{\log_q \frac{1}{1-r}},
$$
and
$$
T(r,f)<\exp_{p}\{(\mu+\varepsilon)\log_q \frac{1}{1-r}\},r\in
E_4,~r\to1^-.
$$
 Moreover, if $p>q\geq1$, then we also have
$$
M(r,f)<\exp_{p+1}\{(\mu+\varepsilon)\log_q \frac{1}{1-r}\},\quad r\in E_4,\;
r\to1^-.
$$
\end{lemma}

\begin{proof}
We use a similar proof as \cite[Lemma 6]{t1}. By the definition of
lower $[p,q]$-order, there exists a
sequence $\{r_n\}_{n=1}^{\infty}$ tending to $1^-$ such that
$1-d(1-r_n)<r_{n+1} (0<d<1)$ (such a sequence
$\{r_n\}_{n=1}^{\infty}$ is called an exponential sequence, see
\cite{d1}), and
$$
\lim_{r_n \to 1^-}{\frac{\log_{p}^+T(r_n,f)}{\log_q
\frac{1}{1-r_n}}}=\mu_{[p,q]}(f).
$$
Then for any $r\in [1-\frac{1-r_n}{d},r_n]$, we have
$$
\frac{\log_{p}^+T(r,f)}{\log_q \frac{1}{1-r}}
\leq\frac{\log_{p}^+T(r_n,f)}{\log_q \frac{1}{1-r_n}}
\frac{\log_q \frac{1}{1-r_n}}{\log_q\frac{1}{1-r}}.
$$
When $q\geq1$, we have $\frac{\log_q \frac{1}{1-r_n}}{\log_q\frac{1}{1-r}}\to1$,
$r_n\to1^-$. Let
$E_4={\bigcup_{n=n_1}^{\infty}}[1-\frac{1-r_n}{d},r_n]$, where $n_1$
is some sufficiently large positive integer, then for any given
$\varepsilon>0$, we have
\begin{gather*}
\lim_{r\to 1^-,\, r\in E_4} \frac{\log_{p}^+T(r,f)}{\log_q \frac{1}{1-r}}
=\lim_{r_n \to 1^-}{\frac{\log_{p}^+T(r_n,f)}{\log_q
\frac{1}{1-r_n}}}=\mu_{[p,q]}(f),
\\
T(r,f)<\exp_{p}\{(\mu+\varepsilon)\log_q \frac{1}{1-r}\},~r\in E_4,~r\to1^-,
\\
\int_{E_4}\frac{dr}{1-r}={\sum_{n=n_1}^{\infty}}
{\int_{1-\frac{1-r_n}{d}}^{r_n}}{\frac{dt}{1-t}}={\sum_{n=n_1}^{\infty}}\log
\frac{1}{d}=\infty.
\end{gather*}
If $p>q\geq1$, then by the standard inequality
$$
T(r,f)\leq\log^+M(r,f)\leq\frac{1+3r}{1-r}T(\frac{1+r}{2},f),
$$
(see e.g. \cite[p. 26]{l1}), we have
$$
\lim_{r\to1^-,\, r\in E_4}
\frac{\log_{p}^+T(r,f)}{\log_q \frac{1}{1-r}}
=\lim_{r\to1^-,\, r\in E_4}
\frac{\log_{p+1}^+M(r,f)}{\log_q \frac{1}{1-r}}.
$$
Therefore,
$$
M(r,f)<\exp_{p+1}\{(\mu+\varepsilon)\log_q \frac{1}{1-r}\},\quad 
r\in E_4,\;r\to 1^-.
$$
\end{proof}

\begin{lemma} \label{lem3.8}
Let $p,q$ be integers such that $p>q\geq 1$,
and $A_{n-1}(z)$, $\dots$, $A_1(z)$, $A_0(z)$ $(\not\equiv 0)$ be analytic
functions in $\Delta$ such that
$\max\{\sigma_{[p,q]}(A_j)|j\neq s\}\leq\mu_{[p,q]}(A_s)\\ <\infty$.
If $f(z)(\not \equiv 0)$ is a solution of \eqref{e1.1}, then we have
$\mu_{[p+1,q]}(f)\leq\mu_{[p,q]}(A_s)$.
\end{lemma}

 \begin{proof}
If $\mu_{[p,q]}(f)<\infty$, then $\mu_{[p+1,q]}(f)=0\leq\mu_{[p,q]}(A_s)$.
So, we assume $\mu_{[p,q]}(f)=\infty$. By Lemma \ref{lem3.1}, we have
\begin{equation}
\begin{aligned}
T(r,f)&=m(r,f)\leq C(\sum_{j=0}^{n-1}\int_0^{2\pi}
 \int_{0}^r|A_j(se^{i\theta})|^{\frac{1}{n-j}}dsd\theta+1)\\
&\leq 2\pi C(\sum_{j=0}^{n-1}rM(r,A_j)+1),
\end{aligned} \label{e3.11}
\end{equation}
where $C=C(n)>0$ is a constant depending on the initial values of $f(z)$
at the point $z_{\theta}$, where $A_j(z_{\theta})\neq 0$ for some
$j=0,\dots,n-1$. Set $b=\max\{\sigma_{[p,q]}(A_j)|j\neq
s\}=\max\{\sigma_{M,[p,q]}(A_j)|j\neq s\}$, then we have
\begin{equation}
M(r,A_j)\leq\exp_{p+1}\{(b+\varepsilon)\log_q\frac{1}{1-r}\},
\quad j\neq s,\label{e3.12}
\end{equation}
for any $\varepsilon>0$ and $r\to1^-$. By Lemma \ref{lem3.7}, there
exists a set $E_4\subset[0,1)$ with
$\int_{E_4}\frac{dr}{1-r}=\infty$ such that
\begin{equation}
M(r,A_s)\leq\exp_{p+1}\{(\mu_{[p,q]}(A_s)+\varepsilon)\log_q\frac{1}{1-r}\},
\quad r\in E_4,\; r\to 1^-.\label{e3.13}
\end{equation}
By \eqref{e3.11}-\eqref{e3.13}, for $r\in E_4$, $r\to 1^-$, we have
\begin{equation}
T(r,f)\leq O\Big(\exp_{p+1}\{(\mu_{[p,q]}(A_s)+2\varepsilon)\log_q\frac{1}{1-r}\}\Big).
\label{e3.14}
\end{equation}
By \eqref{e3.14} and Proposition \ref{prop1.2}, we have
$\mu_{M,[p+1,q]}(f)=\mu_{[p+1,q]}(f)\leq\mu_{[p,q]}(A_s)=\mu_{M,[p,q]}(A_s)$.
\end{proof}

\begin{lemma} \label{lem3.9}
Let $p,q$ be integers such that $p\geq q\geq 1$, and
$A_{n-1}(z)$, $\dots$, $A_1(z)$, $A_0(z)$ $(\not\equiv 0)$ be
analytic functions in $\Delta$. Assume that
$\max\{\sigma_{[p,q]}(A_j)|j=1,\dots,n-1\}\leq\mu_{[p,q]}(A_0)=\mu~(0<\mu<\infty)$
and
$\max\{\tau_{[p,q]}(A_j)|~\sigma_{[p,q]}(A_j)=\mu_{[p,q]}(A_0),j\neq
0\}<\underline{\tau}_{[p,q]}(A_0)=\tau$ $(0<\tau<\infty)$. If
$f(z)(\not\equiv 0)$ is a solution of \eqref{e1.1}, then we have
$\mu_{[p+1,q]}(f)\geq\mu_{[p,q]}(A_0)$.
\end{lemma}

\begin{proof}
Suppose that $f(z)$ is a nonzero solution of \eqref{e1.1}. By \eqref{e1.1}, we get
\begin{equation}
-A_0(z)=\frac{f^{(n)}(z)}{f(z)}+A_{n-1}(z)\frac{f^{(n-1)}(z)}{f(z)}
+\dots+A_1(z)\frac{f'(z)}{f(z)}.\label{e3.15}
\end{equation}
By \eqref{e3.15}, we have
$$
T(r,A_0)=m(r,A_0)\leq\sum_{j=1}^{n-1}m(r,A_j)+\sum_{j=1}^{n}m(r,\frac{f^{(j)}}{f}).
$$
Hence,  by Lemma \ref{lem3.2}, we have
\begin{equation}
T(r,A_0)\leq\sum_{j=1}^{n-1}m(r,A_j)
+O\Big(\log^+ T(r,f)+\log\frac{1}{1-r}\Big),\label{e3.16}
\end{equation}
for $r\not\in E_1$, where $E_1\subset[0,1)$ satisfies
$\int_{E_1}\frac{dt}{1-t}<\infty$. Set
\[
b=\max\{\sigma_{[p,q]}(A_j)|\sigma_{[p,q]}(A_j)
<\mu_{[p,q]}(A_0)=\mu,j=1,\dots,n-1\}.
\]
If $\sigma_{[p,q]}(A_j)<\mu_{[p,q]}(A_0)=\mu$, then for any
$\varepsilon(0<2\varepsilon<\min\{\mu-b,\tau-\tau_1\})$ and all
$r\to1^-$, we have
\begin{equation}
m(r,A_j)= T(r,A_j)\leq\exp_{p}\{(b+\varepsilon)
\log_q\frac{1}{1-r}\}<\exp_{p}\{(\mu-\varepsilon)
\log_q\frac{1}{1-r}\}.\label{e3.17}
\end{equation}
Set $\tau_1=\max\{\tau_{[p,q]}(A_j)|\sigma_{[p,q]}(A_j)=\mu_{[p,q]}(A_0),j\neq
0\}$, then $\tau_1<\tau$. If
$\sigma_{[p,q]}(A_j)=\mu_{[p,q]}(A_0)=\mu$,
$\tau_{[p,q]}(A_j)\leq\tau_1<\tau$, then for $r\to1^-$ and
the above $\varepsilon$, we have
\begin{equation}
m(r,A_j)= T(r,A_j)\leq\exp_{p-1}\big\{(\tau_1+\varepsilon)(\log_{q-1}
\frac{1}{1-r})^{\mu}\big\}.\label{e3.18}
\end{equation}
By the definition of lower $[p,q]$-type, for $r\to1^-$, we
have
\begin{equation}
T(r,A_0)>\exp_{p-1}\big\{(\tau-\varepsilon)(\log_{q-1}\frac{1}{1-r})^{\mu}\big\}.
\label{e3.19}
\end{equation}
By substituting \eqref{e3.17}-\eqref{e3.19} into \eqref{e3.16}, we have
\begin{equation}
\exp_{p-1}\big\{(\tau-2\varepsilon)(\log_{q-1}\frac{1}{1-r})^{\mu}\big\}
\leq O(\log^+ T(r,f)),\quad r\not\in E_1,\; r\to1^-. \label{e3.20}
\end{equation}
Then, by Lemma \ref{lem3.4} and \eqref{e3.20}, for all $r\to 1^-$, we have
$$
\exp_{p-1}\big\{(\tau-2\varepsilon)(\log_{q-1}\frac{1}{1-r})^{\mu}\big\}
\leq O(\log^+ T(s(r),f)),
$$
where $s(r)=1-d(1-r)$, $d\in (0,1)$. Hence, we have
$\mu_{[p+1,q]}(f)\geq\mu_{[p,q]}(A_0)$.
\end{proof}

\begin{lemma} \label{lem3.10}
Let $p,q$ be integers such that $p\geq q\geq 2$ and $f(z)$ be an analytic
function in $\Delta$ with
$0<\sigma_{[p,q]}(f)<\infty$. Then for any given $\varepsilon>0$,
there exists a set $E_5\subset[0,1)$ with
$\int_{E_5}\frac{dr}{1-r}=\infty$ such that
$$
\tau=\tau_{[p,q]}(f)=\lim_{r\to1^-,\,r\in E_5}
\frac{\log_{p-1}^+T(r,f)}{(\log_{q-1} \frac{1}{1-r})^{\sigma_{[p,q]}(f)}}.
$$
\end{lemma}

\begin{proof}
By the definition of $[p,q]$-type, there exists a
sequence $\{r_n\}_{n=1}^{\infty}$ tending to $1^-$ satisfying
$1-d(1-r_n)<r_{n+1} (0<d<1)$ such that
$$
\tau_{[p,q]}(f)=\lim_{{r_n\to1^-}}\frac{\log_{p-1}
^+T(r_n,f)}{(\log_{q-1} \frac{1}{1-r_n})^{\sigma_{[p,q]}(f)}}.
$$
Then for any $r\in [r_n,1-d(1-r_n)]$, we have
$$
\frac{\log_{p-1}^+T(r_n,f)}{(\log_{q-1}\frac{1}{1-r_n})^{\sigma_{[p,q]}(f)}}
\Big(\frac{\log_{q-1}\frac{1}{1-r_n}}{\log_{q-1}\frac{1}{1-r}}\Big)^{\sigma_{[p,q]}(f)}
\leq \frac{\log_{p-1} ^+T(r,f)}{(\log_{q-1}\frac{1}{1-r})^{\sigma_{[p,q]}(f)}}.
$$ 
When $q\geq2$, we have
\[
\frac{\log_{q-1}\frac{1}{1-r_n}}{\log_{q-1}\frac{1}{1-r}}\to1,\quad
r_n\to1^-.
\]
Let
$E_5=\bigcup_{n=n_1}^{\infty}[r_n,1-d(1-r_n)]$, where $n_1$ is some
sufficiently large positive integer, then we have
$$
\lim_{r\to1^-,\,r\in E_5} \frac{\log_{p-1}^+T(r,f)}{(\log_{q-1} 
\frac{1}{1-r})^{\sigma_{[p,q]}(f)}}
=\lim_{r_n \to1^-} {\frac{\log_{p-1}^+T(r_n,f)}{(\log_{q-1}
\frac{1}{1-r_n})^{\sigma_{[p,q]}(f)}}}=\tau_{[p,q]}(f),
$$ 
and
\[
\int_{E_5}\frac{dr}{1-r}={\sum_{n=n_1}^{\infty}}{\int_{r_n}^{1-d(1-r_n)}}
{\frac{dt}{1-t}}={\sum_{n=n_1}^{\infty}}\log\frac{1}{d}=\infty.
\]
\end{proof}

\begin{lemma} \label{lem3.11}
Let $p,q$ be integers such that $p> q\geq 1$.
If $A_{n-1}(z)$, \dots, $A_1(z)$, $A_0(z)$ $(\not\equiv 0)$ are analytic
functions of $[p,q]$-order in $\Delta$, then every solution
$f(z)(\not\equiv 0)$ of \eqref{e1.1} satisfies
$\sigma_{[p+1,q]}(f)\leq\max\{\sigma_{[p,q]}(A_j)|j=0,\dots,n-1\}$.
\end{lemma}

\begin{proof}
Set
$b=\max\{\sigma_{[p,q]}(A_j)|j=0,\dots,n-1\}
=\max\{\sigma_{M,[p,q]}(A_j)|j=0,\dots,n-1\}$.
Then we have
\begin{equation}
M(r,A_j)\leq\exp_{p+1}\{(b+\varepsilon)\log_q\frac{1}{1-r}\},\label{e3.21}
\end{equation}
for any given $\varepsilon>0$ and $r\to1^-$. By \eqref{e3.11} and
\eqref{e3.21}, for the above $\varepsilon>0$ and $r\to1^-$, we have
\begin{equation}
T(r,f)=m(r,f)\leq O(\exp_{p+1}\{(b+2\varepsilon)\log_q\frac{1}{1-r}\}).\label{e3.22}
\end{equation}
Therefore,
$\sigma_{[p+1,q]}(f)\leq\max\{\sigma_{[p,q]}(A_j)|j=0,\dots,n-1\}$.
\end{proof}

\section{Proofs of main theorems}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
By Lemma \ref{lem3.11}, we have
$\sigma_{[p+1,q]}(f)\leq\sigma_{[p,q]}(A_0)$. 
Set
$b=\max\{\sigma_{[p,q]}(A_j)|\sigma_{[p,q]}(A_j)<\sigma_{[p,q]}(A_0)\}$.
If $\sigma_{[p,q]}(A_j)<\mu_{[p,q]}(A_0)\leq\sigma_{[p,q]}(A_0)$ or
$\sigma_{[p,q]}(A_j)\leq\mu_{[p,q]}(A_0)<\sigma_{[p,q]}(A_0)$, then
for any given $\varepsilon(0<2\varepsilon<\sigma_{[p,q]}(A_0)-b)$
and $r\to1^-$, we have
\begin{equation}
\begin{aligned}
m(r,A_j)&= T(r,A_j)\leq\exp_{p}\big\{(b+\varepsilon)\log_q\frac{1}{1-r}\big\}\\
& <\exp_{p}\big\{(\sigma_{[p,q]}(A_0)-\varepsilon)\log_q\frac{1}{1-r}\big\}.
\end{aligned} \label{e4.1}
\end{equation}
Set
$\tau_1=\max\{\tau_{[p,q]}(A_j)|~\sigma_{[p,q]}(A_j)=\mu_{[p,q]}(A_0),j\neq
0\}$. If $\sigma_{[p,q]}(A_j)=\mu_{[p,q]}(A_0)=\sigma_{[p,q]}(A_0)$,
then we have $\tau_1<\tau\leq\tau_{[p,q]}(A_0)$. Therefore,
\begin{equation}
m(r,A_j)= T(r,A_j)\leq\exp_{p-1}
\big\{(\tau_1+\varepsilon)(\log_{q-1}\frac{1}{1-r})^{\sigma_{[p,q]}(A_0)}\big\}
\label{e4.2}
\end{equation}
holds for $r\to1^-$ and any given
$\varepsilon~(0<2\varepsilon<\tau_{[p,q]}(A_0)-\tau_1)$. By the
definition of $[p,q]$-type and Lemma \ref{lem3.10}, for all
$r\to1^-,\,r\in E_5$, where $E_5\subset[0,1)$ satisfies
$\int_{E_5}\frac{dr}{1-r}=\infty$, we have
\begin{equation}
T(r,A_0)>\exp_{p-1}\big\{(\tau_{[p,q]}(A_0)-\varepsilon)
(\log_{q-1}\frac{1}{1-r})^{\sigma_{[p,q]}(A_0)}\big\}.\label{e4.3}
\end{equation}
Then by \eqref{e3.16} and \eqref{e4.1}-\eqref{e4.3}, for all
$r\to1^-,\,r\in E_5\backslash E_1$ and the above $\varepsilon$, where
$E_1\subset[0,1)$ satisfies $\int_{E_1}\frac{dt}{1-t}<\infty$, we
have
\begin{equation}
\exp_{p-1}\big\{(\tau_{[p,q]}(A_0)-2\varepsilon)(\log_{q-1}
\frac{1}{1-r})^{\sigma_{[p,q]}(A_0)}\big\}\leq O(\log^+ T(r,f)).\label{e4.4}
\end{equation}
By \eqref{e4.4}, $\sigma_{[p+1,q]}(f)\geq\sigma_{[p,q]}(A_0)$.
Thus, we have $\sigma_{[p+1,q]}(f)=\sigma_{[p,q]}(A_0)$.

By Lemmas \ref{lem3.8} and \ref{lem3.9}, we know that every solution 
$f(z)$ $(\not\equiv 0)$ of \eqref{e1.1} satisfies 
$\mu_{[p+1,q]}(f)=\mu_{[p,q]}(A_0)$.

Now,  we need to prove
$\overline{\underline{\lambda}}_{[p+1,q]}(f-\varphi)=\mu_{[p+1,q]}(f)$
and $\overline{\lambda}_{[p+1,q]}(f-\varphi)=\sigma_{[p+1,q]}(f)$.
 Setting $g=f-\varphi$, since $\sigma_{[p+1,q]}(\varphi)<\mu_{[p,q]}(A_0)$,
we have
$\sigma_{[p+1,q]}(g)=\sigma_{[p+1,q]}(f)=\sigma_{[p,q]}(A_0)$,
$\mu_{[p+1,q]}(g)=\mu_{[p+1,q]}(f)=\mu_{[p,q]}(A_0)$,
$\overline{\lambda}_{[p+1,q]}(g)=\\ \overline{\lambda}_{[p+1,q]}(f-\varphi)$
and
$\overline{\underline{\lambda}}_{[p+1,q]}(g)
=\overline{\underline{\lambda}}_{[p+1,q]}(f-\varphi)$.
By substituting $f=g+\varphi, f'=g'+\varphi',\dots,
f^{(n)}=g^{(n)}+\varphi^{(n)}$ in \eqref{e1.1}, we obtain
\begin{equation}
g^{(n)}+A_{n-1}(z)g^{(n-1)}+\dots+A_0(z)g
=-[\varphi^{(n)}+A_{n-1}(z)\varphi^{(n-1)}+\dots+A_0(z)\varphi].\label{e4.5}
\end{equation}
If
$F(z)=\varphi^{(n)}+A_{n-1}(z)\varphi^{(n-1)}+\dots+A_0(z)\varphi\equiv 0$,
then by Lemma \ref{lem3.9}, we have
$\mu_{[p+1,q]}(\varphi)\geq\mu_{[p,q]}(A_0)$, which is a
contradiction. Thus, $F(z)\not\equiv 0$. Since
$\sigma_{[p+1,q]}(F)\leq\sigma_{[p+1,q]}(\varphi)<\mu_{[p,q]}(A_0)
=\mu_{[p+1,q]}(f)=\mu_{[p+1,q]}(g)\leq\sigma_{[p+1,q]}(g)=\sigma_{[p+1,q]}(f)$,
by Lemma \ref{lem3.5} and \eqref{e4.5}, we have $\overline{\lambda}_{[p+1,q]}(g)$
$=\lambda_{[p+1,q]}(g)=\sigma_{[p+1,q]}(g)=\sigma_{[p,q]}(A_0)$;
i.e.,
$\overline{\lambda}_{[p+1,q]}(f-\varphi)
=\lambda_{[p+1,q]}(f-\varphi)=\sigma_{[p+1,q]}(f)=\sigma_{[p,q]}(A_0)$.
By Lemma \ref{lem3.6} and \eqref{e4.5}, we have
$\overline{\underline{\lambda}}_{[p+1,q]}(g)=\mu_{[p+1,q]}(g)$; i.e.,
$\overline{\underline{\lambda}}_{[p+1,q]}(f-\varphi)
=\mu_{[p+1,q]}(f)=\mu_{[p,q]}(A_0)$.
Therefore,
$\overline{\underline{\lambda}}_{[p+1,q]}(f-\varphi)
=\mu_{[p+1,q]}(f)=\mu_{[p,q]}(A_0)\leq\sigma_{[p,q]}(A_0)
=\sigma_{[p+1,q]}(f)=\overline{\lambda}_{[p+1,q]}(f-\varphi)
={\lambda}_{[p+1,q]}(f-\varphi)$.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.2}]
By Lemma \ref{lem3.11}, we obtain
$\sigma_{[p+1,q]}(f)\leq\sigma_{[p,q]}(A_0)$. By
\begin{equation}
\limsup_{r\to1^-} {\sum_{j=1}^{n-1}m(r,A_j)}/{m(r,A_0)}<1, \label{e4.6}
\end{equation}
for $r\to1^-$, we have
\begin{equation}
\sum_{j=1}^{n-1}m(r,A_j)<\delta m(r,A_0)=\delta T(r,A_0),\label{e4.7}
\end{equation}
 where $\delta\in (0,1)$. By \eqref{e3.16} and
\eqref{e4.7}, for $r\to1^-,\, r\not\in  E_1$, where
$E_1\subset[0,1)$ satisfies $\int_{E_1}\frac{dt}{1-t}<\infty$, we have
\begin{equation}
T(r,A_0)\leq O(\log^+ T(r,f)+\log\frac{1}{1-r}).\label{e4.8}
\end{equation}
By Lemma \ref{lem3.4} and \eqref{e4.8}, we have
$\sigma_{[p+1,q]}(f)\geq\sigma_{[p,q]}(A_0)$. Thus, 
$\sigma_{[p+1,q]}(f)=\sigma_{[p,q]}(A_0)$.

By Lemma \ref{lem3.4} and \eqref{e4.8}, we have
$\mu_{[p+1,q]}(f)\geq\mu_{[p,q]}(A_0)$. By Lemma \ref{lem3.8}, we have
$\mu_{[p+1,q]}(f)\leq\mu_{[p,q]}(A_0)$. Thus, 
$\mu_{[p+1,q]}(f)=\mu_{[p,q]}(A_0)$.

Using a proof similar to the one in Theorem \ref{thm2.1}, we obtain
$\overline{\underline{\lambda}}_{[p+1,q]}(f-\varphi)
=\mu_{[p+1,q]}(f)=\mu_{[p,q]}(A_0)\leq\sigma_{[p,q]}(A_0)
=\sigma_{[p+1,q]}(f)=\overline{\lambda}_{[p+1,q]}(f-\varphi)
={\lambda}_{[p+1,q]}(f-\varphi)$.
The proof is complete.
\end{proof}

\subsection*{Acknowledgements}
We want to thank the anonymous referees and the editors for
their comments and suggestions.

This research was supported by the National Natural Science
Foundation of China (11301233, 11171119), the Youth Science
Foundation of Education Bureau of Jiangxi Province in China
(GJJ14271), and Sponsored Program for Cultivating Youths of
Outstanding Ability in Jiangxi Normal University of China.

\begin{thebibliography}{99}

\bibitem{b1} S. Bank;
\emph{General Theorem Concerning the Growth of Solutions of First-order 
 Algebraic Differential Equations}, Composito Math., 25, 1972, 61-70.

\bibitem{b2} B. Bela\"{\i}di;
\emph{Growth of Solutions to Linear Differential Equations with Analytic 
Coefficients of $[p,q]$-Order in the Unit Disc}, 
Electron. J. Diff. Equ., 156, 2011, 1-11.

\bibitem{b3} B. Bela\"{\i}di;
\emph{Growth and Oscillation Theory of $[p,q]$-order Analytic Solutions 
of Linear Differential Equations in the Unit Disc}, 
J. of Math. Analysis, 3, 2012, 1-11.

\bibitem{c1} T. B. Cao, H. X. Yi;
\emph{The Growth of Solutions of Linear Differential Equations with 
Coeffficients of Iterated Order in the Unit Disc}, 
J. Math. Anal. Appl., 319(1), 2006, 278-294.

\bibitem{c2} T. B. Cao;
\emph{The Growth, Oscillation and Fixed Points of
Solutions of Complex Linear Differential Equations in the Unit Disc},
J. Math. Anal. Appl., 352, 2009, 739-748.

\bibitem{d1} P. Duren, Theory of Hp Spaces. Academic Press, New York-San Francisco-London, 1970.

\bibitem{h1} W.K. Hayman;
\emph{Meromorphic Functions}, Oxford: Clarendon Press, 1964.

\bibitem{h2} J. Heittokangas;
\emph{On Complex Differential Equations in the Unit Disc}, 
Ann. Acad.Sci. Fenn. Math. Diss., 122, 2000, 1-54.

\bibitem{h3} J. Heittokangas, R. Korhonen, J. R\"{a}tty\"{a};
\emph{Growth Estimates for Solutions of Linear Complex Differential Equations}, 
Ann. Acad. Sci. Fenn. Math., 29(1), 2004, 233-246.

\bibitem{h4} H. Hu;
\emph{Growth and Value Distribution of the Solutions
of  Homogeneous Linear Differential Equations in the Complex Plane
and the Unit Disc}, the Master Degree Thesis, Jiangxi Normal
University, 2013, 1-44.

\bibitem{j1} O. P. Juneja, G. P. Kapoor, S. K. Bajpai;
\emph{On the $[p,q]$-Order and Lower $[p,q]$-Order of an Entire Function}, 
J. Reine Angew. Math., 282, 1976, 53-67.

\bibitem{j2} O. P. Juneja, G. P. Kapoor, S. K. Bajpai;
\emph{On the $[p,q]$-Type and Lower $[p,q]$-Type of an Entire Function}, 
J. Reine Angew. Math., 290, 1977, 180-190.

\bibitem{l1} I. Laine;
\emph{Nevanlinna Theory and Complex Differential Equations}, 
Berlin: Walter de Gruyter, 1993.

\bibitem{l2} I. Laine;
\emph{Complex Differential Equations, Handbook of Differential Equations}: 
Ordinary Differential Equations, 4, 2008, 269-363.

\bibitem{l3} J. Liu, J. Tu, L. Z. Shi;
\emph{Linear Differential Equations with Entire Coefficients of $[p,q]$-Order
 in the Complex Plane}, J. Math. Anal. Appl. 372, 2010, 55-67.

\bibitem{s1} D. F. Shea, L. R. Sons;
\emph{Value Distibution Theory for Mermorphic Solutions of Slow Growth 
in the Disk}, Houston J. Math., 12 (2), 1986, 249-266.

\bibitem{t1} J. Tu, G.T. Deng;
\emph{Growth of Solutions of Higher Order Linear
Differential Equations with the Coefficient $A_0$ Being Dominant},
Acta Math. Sci., 30A(4), 2010, 945-952.

\end{thebibliography}

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