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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 156, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/156\hfil Growth of solutions]
{Growth of solutions to linear differential equations with
analytic coefficients of [p,q]-order in the unit disc}

\author[B. Bela\"idi\hfil EJDE-2011/156\hfilneg]
{Benharrat Bela\"idi}

\address{Department of Mathematics\\
Laboratory of Pure and Applied Mathematics\\
University of Mostaganem (UMAB)\\
B. P. 227 Mostaganem, Algeria}
\email{belaidi@univ-mosta.dz}

\thanks{Submitted August 4, 2011. Published November 18, 2011.}
\subjclass[2000]{34M10, 30D35}
\keywords{Linear differential equations; analytic function, $[p,q]$-order}

\begin{abstract}
 In this article, we study the growth of solutions to
 complex higher-order linear differential equations in
 which the coefficients are analytic functions of
 $[p,q]$-order in the unit disc.
\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 and statement of results}

 For $k\geq 2$ we consider the linear differential
equations
\begin{gather}
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots +A_1(z)f'+A_0(z)f=0,  \label{e1.1}
\\
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots +A_1(z)f'+A_0(z)f=F(z),  \label{e1.2}
\end{gather}
where $A_0(z),\dots ,A_{k-1}(z)$, $F(z)\not\equiv 0$
are analytic functions in the unit disc
$\Delta =\{z\in \mathbb{C}:|z|<1\}$. It is well-known that all
solutions of \eqref{e1.1} and \eqref{e1.2} are analytic functions
in $\Delta $ and that there are exactly $k$ linearly independent
solutions of \eqref{e1.1} (see \cite{h2}).
Juneja, Kapoor and Bajpai \cite{j1,j2} have investigated
properties of entire functions of $[p,q]$-order and obtained
some results.
 Liu, Tu and Shi \cite{l4}, by using the concept of $[p,q] $-order
have considered equations \eqref{e1.1}, \eqref{e1.2} with entire
coefficients and obtained different results
concerning the growth of its solutions. Recently, there has been an
increasing interest in studying the growth of analytic solutions
of linear differential equations in the unit disc by making
use of Nevanlinna theory
(see \cite{b2,b3,c1,c2,c3,f1,h2,h3,l3}).
In this article, we continue to consider this
subject and investigate the complex linear differential equations
\eqref{e1.1} and \eqref{e1.2} when the coefficients
$A_0,A_1,\dots ,A_{k-1}$, $F$ are analytic functions of
$[p,q]$-order in $\Delta $.


 In this article, we assume that the reader is
familiar with the fundamental results and the standard notation
 of the Nevanlinna's theory in the unit disc
$\Delta =\{z\in\mathbb{C}:|z|<1\}$ (see \cite{h1,h2,l2,t1}).

Before, we state our results we need to give some
definitions and discussions. Firstly, let us give definition about the
degree of small growth order of functions in $\Delta $ as polynomials on the
complex plane $\mathbb{C}$.
There are many  definitions of small growth order of functions in
$\Delta $; see \cite{c3,c4}.


 \begin{definition} \label{def1.1}\rm
 For a meromorphic function $f$ in $\Delta $ let
\begin{equation*}
D(f):=\limsup_{r\to 1^{-}}\frac{T(r,f)}{\log \frac{1}{1-r}},
\end{equation*}
where $T(r,f)$ is the Nevanlinna characteristic function of $f$.
If $D(f)<\infty $, we say that $f$ is of finite degree
 $D( f)$ (or is non-admissible); if $D(f)=\infty $, we say
that $f$ is of infinite degree (or is admissible).
If $f$ is an analytic function in $\Delta $, and
\begin{equation*}
D_M(f):=\limsup_{r\to 1^{-}} \frac{\log ^{+}M(r,f)}
{\log \frac{1}{1-r}}
\end{equation*}
in which $M(r,f)=\max_{|z|=r} |f(z)|$ is the maximum modulus
function, then we say that $f$ is a function of finite degree
$D_M(f)$ if $D_M(f)<\infty $; otherwise, $f$ is of infinite
degree.
\end{definition}

 Now, we give the definitions of iterated order and growth
index to classify generally the functions of fast growth
in $\Delta $ as those in $\mathbb{C}$ ;
see \cite{b4,k1,l1}.
Let us define inductively, for $r\in [0,1)$,
$\exp_1r:=e^{r}$ and $\exp_{p+1}r:=\exp (\exp_pr)$,
$p\in\mathbf{\mathbb{N}}$.
We also define for all $r$ sufficiently large in $(0,1)$,
$\log_1r:=\log r$ and $\log_{p+1}r:=\log (\log_pr)$,
$p\in \mathbf{\mathbb{N}}$. Moreover, we denote by
$\exp_0r:=r$, $\log_0r:=r$, $\log_{-1}r:=\exp_1r$ and
$\exp_{-1}r:=\log_1r$.


 \begin{definition}[\cite{c1,c2,l2}] \label{def1.2} \rm
Let $f$ be a meromorphic function in $\Delta $. Then the iterated
$p$-order of $f$ is defined by
\begin{equation*}
\rho_p(f)=\limsup_{r\to 1^{-}}
\frac{\log_p^{+}T(r, f)}{\log \frac{1}{1-r}}
\quad (p\text{ is an integer, }p\geq 1),
\end{equation*}
where $\log_1^{+}x=\log ^{+}x=\max \{\log x,0\}$,
$\log_{p+1}^{+}x=\log ^{+}\log_p^{+}x$. For $p=1$, this notation
is called order and for $p=2$ hyper-order \cite{h2,l3}.
If $f$ is analytic in $\Delta $, then the iterated $p$-order of
$f$ is defined by
\begin{equation*}
\rho_{M,p}(f)=\limsup_{r\to 1^{-}}
\frac{\log_{p+1}^{+}M(r, f)}{\log \frac{1}{1-r}}
\quad (p\text{ is an integer, }p\geq 1).
\end{equation*}
\end{definition}

\begin{remark} \label{rmk1.1} \rm
It follows by Tsuji  \cite[p. 205]{t1} that if $f$ is
an analytic function in $\Delta $, then we have the inequalities
\begin{equation*}
\rho_1(f)\leq \rho_{M,1}(f)\leq \rho_1(f)+1
\end{equation*}
which are the best possible in the sense that there are analytic
functions $g $ and $h$ such that $\rho_{M,1}(g)=\rho_1(g)$
and $\rho_{M,1}(h)=\rho_1(h)+1$, see \cite{c4}.
However, it follows by \cite[Proposition 2.2.2]{l1}
that $\rho_{M,p}(f)=\rho_p(f)$ for $p\geq  2$.
\end{remark}

\begin{definition}[\cite{c1}] \label{def1.3}\rm
The growth index of the iterated order of a meromorphic function
$f(z)$ in $\Delta $ is defined by
\begin{equation*}
i(f)=\begin{cases}
0,& \text{if $f$ is non-admissible,} \\
\min \{j\in \mathbf{\mathbb{N}}:\rho_{j}(f)<\infty \}
& \text{if $f$ is admissible,} \\
\infty , & \text{if }\rho_{j}(f)=\infty\text{ for all }
j\in \mathbf{\mathbb{N}}.
\end{cases}
\end{equation*}
For an analytic function $f$ in $\Delta $, we also define
\begin{equation*}
i_M(f)=\begin{cases}
0,&\text{if $f$ is non-admissible,}\\
\min \{j\in \mathbf{\mathbb{N}}:\rho_{M,j}(f)<\infty \}
&\text{if $f$ is admissible,} \\
\infty ,& \text{if }\rho_{M,j}(f)=\infty \text{ for all }
j\in \mathbf{\mathbb{N}}.
\end{cases}
\end{equation*}
\end{definition}

\begin{remark} \label{rmk1.2} \rm
If $\rho_p(f)$ $<\infty $ or $i(f)\leq p$, then
we say that $f$ is of finite iterated $p$-order;
if $\rho_p(f)$ $=\infty $ or $i(f)>p$, then we say that $f$
is of infinite iterated $p$-order. In particular, we say that
$f$ is of finite order if $\rho_1(f)$ $<\infty $
or $i(f)\leq 1$; $f$ is of infinite order if $\rho_1(f)=\infty $
or $i(f)>1$.
\end{remark}


 Now, we introduce the concept of $[p,q]$-order for
meromorphic and analytic functions in the unit disc.


 \begin{definition} \label{def1.4} \rm
Let $p\geq  q\geq  1$ be integers. Let $f$ be meromorphic function
in $\Delta $, the $[p,q]$-order of $f(z)$ is defined by
\begin{equation*}
\rho_{[p,q] }(f)=\limsup_{r\to 1^{-}}
\frac{\log_p^{+}T(r,f)}{\log_q\frac{1}{1-r}}.
\end{equation*}
For an analytic function $f$ in $\Delta $, we also define
\begin{equation*}
\rho_{M,[p,q] }(f)=\limsup_{r\to 1^{-}}
\frac{\log_{p+1}^{+}M(r,f)}{\log_q\frac{1}{1-r}}.
\end{equation*}
\end{definition}

\begin{remark} \label{rmk1.3} \rm
It is easy to see that $0\leq \rho_{[p,q] }(f)\leq \infty $.
If $f(z)$ is non-admissible, then $\rho_{[p,q] }(f)=0$ for any
$p\geq  q\geq  1$. By Definition \ref{def1.4},
we have that $\rho_{[1,1] }(f)=\rho_1(f)=\rho (f)$,
$\rho_{[2,1] }(f)=\rho_2(f)$
and $\rho_{[p+1,1] }(f)=\rho_{p+1}(f)$.
\end{remark}

\begin{proposition} \label{prop1.1}
Let $p\geq  q\geq  1$ be integers, and let $f$ be analytic
function in $\Delta $ of $[p,q] $-order. The
following two statements hold:
\begin{itemize}
\item[(i)]  If $p=q$, then
\begin{equation*}
\rho_{[p,q] }(f)\leq \rho_{M,[p,q
] }(f)\leq \rho_{[p,q] }(f)+1.
\end{equation*}

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

\begin{proof}
By the standard inequalities \cite[p. 26]{l1}
\begin{equation*}
T(r,f)\leq \log ^{+}M(r,f)\leq \frac{1+3r
}{1-r}T(\frac{1+r}{2},f),
\end{equation*}
we easily deduce that (i) and (ii) hold.
\end{proof}

 The present article may be understood as an extension and
improvement of the recent article of the author \cite{b3}. We
obtain the following results.

\begin{theorem} \label{thm1.1}
Let $p\geq  q\geq  1$ be integers,  and let
$A_0(z),\dots ,A_{k-1}(z)$ be analytic functions in the unit disc
$\Delta $. Suppose that there exists a sequence of complex numbers
$(z_n)_{n\in\mathbb{N}}$ with
$|z_n|=r_n\to 1^{-}$, $n\to \infty $ such that for real constants
$\alpha$, $\beta $ where $0\leq \beta <\alpha $,
we have
\begin{equation}
T(r_n,A_0)\geq  \exp_p\{\alpha \log_q(
\frac{1}{1-r_n})\}\label{e1.3}
\end{equation}
as $n\to \infty $, and
\begin{equation}
T(r,A_{j})\leq \exp_p\{\beta \log_q(
\frac{1}{1-r})\}\quad (j=1,\dots ,k-1)\label{e1.4}
\end{equation}
holds for all $r\in [0,1)$. Then every solution
$f\not\equiv 0$ of  \eqref{e1.1} satisfies
$\rho_{[p,q] }(f)=\rho_{M,[p,q]}(f)=\infty $ and
$\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q] }(f)\geq  \alpha $.
\end{theorem}


\begin{theorem} \label{thm1.2}
Let $p\geq  q\geq  1$ be integers, and let
$A_0(z),\dots ,A_{k-1}(z)$
be analytic functions in the unit disc $\Delta $ satisfying
$\max \{\rho_{[p,q] }(A_{j}):j=1,\dots ,k-1\}
\leq \rho_{[p,q] }(A_0)=\rho $.
Suppose that there exist a sequence of complex numbers
$(z_n)_{n\in\mathbb{N}}$ with
$|z_n|=r_n\to 1^{-}$,  $n\to \infty $  and a real number
$\mu $ satisfying $0\leq \mu <\rho $ such that for any given
$\varepsilon $ $(0<\varepsilon <\rho -\mu )$
sufficiently small, we have
\begin{equation}
T(r_n,A_0)\geq  \exp_p\{(\rho
-\varepsilon )\log_q(\frac{1}{1-r_n})\}
\label{e1.5}
\end{equation}
as $n\to \infty $, and
\begin{equation}
T(r,A_{j})\leq \exp_p\{\mu \log_q(\frac{1
}{1-r})\}\quad (j=1,\dots ,k-1)\label{e1.6}
\end{equation}
holds for all $r\in [0,1)$. Then every solution
 $f\not\equiv 0$ of  \eqref{e1.1} satisfies
$\rho_{[p,q] }(f)=\rho_{M,[p,q]}(f)=\infty $  and
\begin{equation*}
\rho_{[p,q] }(A_0)\leq \rho_{[p+1,q
] }(f)=\rho_{M,[p+1,q] }(f)
\leq \max \{\rho_{M,[p,q] }(A_{j})
:j=0,1,\dots ,k-1\}.
\end{equation*}
Furthermore, if $p>q$, then
\begin{equation*}
\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q]}(f)=\rho_{[p,q] }(A_0).
\end{equation*}
\end{theorem}

\begin{theorem} \label{thm1.3}
Let $p\geq  q\geq  1$  be integers.
Let $A_0(z),\dots$, $A_{k-1}(z)$  and
$F(z)\not\equiv 0$ be analytic functions in
the unit disc $\Delta $ such that for some integer
$s$, $1\leq s\leq k-1$ satisfying
$\max \{\rho_{[p,q] }(A_{j})$ $(j\neq s)$,
$\rho_{[p,q] }(F)\}<\rho_{[p,q] }(A_{s})$.
Then every admissible solution $f$  of \eqref{e1.2}
with $\rho_{[p,q]}(f)<\infty $ satisfies
$\rho_{[p,q]}(f)\geq  \rho_{[p,q] }(A_{s})$.
\end{theorem}


\begin{theorem} \label{thm1.4}
Let $p\geq  q\geq  1$  be integers. Let $A_0(z),\dots$,
$A_{k-1}(z)$ and $F(z)\not\equiv 0$ be analytic functions
in the unit disc $\Delta $ such that for
some integer $s$, $0\leq s\leq k-1$, we have
$\rho_{[p,q] }(A_{s})=\infty $ and
$\max \{\rho_{[p,q] }(A_{j})\; (j\neq s),\rho_{[p,q] }(F)\}<\infty$.
Then every solution $f$ of \eqref{e1.2} satisfies
$\rho_{[p,q]}(f)=\infty $.
\end{theorem}

\section{Preliminaries}

 In this section we give some lemmas which are used in the
proofs of our theorems.


\begin{lemma}[\cite{h2}] \label{lem2.1}
 Let $f$ be a meromorphic function in the unit disc $\Delta $,
 and let $k\geq  1$\ be an integer. Then
\begin{equation}
m\big(r,\frac{f^{(k)}}{f}\big)=S(r,f),
\label{e2.1}
\end{equation}
where $S(r,f)=O\big(\log ^{+}T(r,f)+\log (\frac{1}{1-r})\big)$,
possibly outside a set $E_1\subset [0,1)$ with
 $\int_{E_1}\frac{dr}{1-r}<\infty $.
\end{lemma}

 Next we give the generalized logarithmic derivative
lemma.


\begin{lemma} \label{lem2.2}
Let $p\geq  q\geq  1$ be integers. Let $f$ be a meromorphic
function in the unit disc $\Delta $ such that
$\rho_{[p,q] }(f)=\rho <\infty $, and let
$k\geq  1$\ be an integer. Then for any $\varepsilon >0$,
\begin{equation}
m\big(r,\frac{f^{(k)}}{f}\big)=O\big(\exp_{p-1}\{
(\rho +\varepsilon )\log_q(\frac{1}{1-r})
\}\big)\label{e2.2}
\end{equation}
holds for all $r$ outside a set $E_2\subset [0,1)$
with $\int_{E_2}\frac{dr}{1-r}<\infty $.
\end{lemma}

\begin{proof}
First for $k=1$. Since $\rho_{[p,q]}(f)=\rho <\infty $,
for all $r\to 1^{-}$ we have
\begin{equation}
T(r,f)\leq \exp_p\{(\rho +\varepsilon
)\log_q(\frac{1}{1-r})\}.  \label{e2.3}
\end{equation}
By Lemma \ref{lem2.1}, we have
\begin{equation}
m\big(r,\frac{f'}{f}\big)=O\big(\ln ^{+}T(r,f)
+\ln (\frac{1}{1-r})\big)\label{e2.4}
\end{equation}
holds for all $r$ outside a set $E_2\subset [0,1)$ with
$\int_{E_2}\frac{dr}{1-r}<\infty $. Hence, we obtain
\begin{equation}
m(r,\frac{f'}{f})=O(\exp_{p-1}\{(
\rho +\varepsilon )\log_q(\frac{1}{1-r})\}
), r\notin E_2.  \label{e2.5}
\end{equation}
Next, we assume that we have
\begin{equation}
m\big(r,\frac{f^{(k)}}{f}\big)=O\big(\exp_{p-1}\{
(\rho +\varepsilon )\log_q(\frac{1}{1-r})
\}\big), \quad r\notin E_2  \label{e2.6}
\end{equation}
for some an integer $k\geq  1$. Since $N(r,f^{(k)})\leq (k+1)N(r,f)$,
it holds that
\begin{equation}
\begin{split}
T(r,f^{(k)})&=m(r,f^{(k)}) +N(r,f^{(k)}) \\
&\leq m\big(r,\frac{f^{(k)}}{f}\big)+m(r,f)+(k+1)N(r,f) \\
&\leq m\big(r,\frac{f^{(k)}}{f}\big)+(k+1)
T(r,f)=O\big(\exp_{p-1}\{(\rho +\varepsilon
)\log_q(\frac{1}{1-r})\}\big) \\
&\quad +(k+1)T(r,f)=O\big(\exp_p\{(\rho
+\varepsilon )\log_q(\frac{1}{1-r})\}\big).
\end{split}\label{e2.7}
\end{equation}
By \eqref{e2.4} and \eqref{e2.7}, we obtain
\begin{equation}
m\big(r,\frac{f^{(k+1)}}{f^{(k)}}\big)
=O\big(\exp_{p-1}\{(\rho +\varepsilon )\log_q(
\frac{1}{1-r})\}\big), \quad r\notin E_2
 \label{e2.8}
\end{equation}
and hence,
\begin{equation}
\begin{split}
m\big(r,\frac{f^{(k+1)}}{f}\big)
&\leq m\big(r,\frac{ f^{(k+1)}}{f^{(k)}}\big)
+m\big(r,\frac{ f^{(k)}}{f}\big)\\
&=O\big(\exp_{p-1}\{(\rho +\varepsilon )\log_q(
\frac{1}{1-r})\}\big), \quad r\notin E_2.
\end{split} \label{e2.9}
\end{equation}
\end{proof}

\begin{lemma}[\cite{b1}] \label{lem2.3}
Let $g:(0,1)\to \mathbf{\mathbb{R}}$ and
$h:(0,1)\to \mathbf{\mathbb{R}}$ be monotone increasing
functions such that $g(r)\leq h(r)$ holds outside of an exceptional
set $E_3\subset [0,1)$ for which
$\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}[\cite{h4}] \label{lem2.4}
Let $f$ be a solution of equation \eqref{e1.1}, where
the coefficients $A_{j}(z)$ $(j=0,\dots ,k-1)$ are analytic
functions in the disc $\Delta_{R}=\{z\in\mathbb{C}:|z|<R\}$,
$0<R\leq \infty $.
Let $n_{c}\in \{1,\dots ,k\}$\ be the number of
nonzero coefficients $A_{j}(z)$ $(j=0,\dots ,k-1)$,
 and let $\theta \in [0,2\pi ] $
 and $\varepsilon >0$. If $z_{\theta }=\nu e^{i\theta}\in \Delta_{R}$
 is such that $A_{j}(z_{\theta })\neq 0$ for some
$j=0,\dots ,k-1$, then for all $\nu <r<R$,
\begin{equation}
|f(re^{i\theta })|\leq C\exp \Big(
n_{c} \int_\nu^r \max_{j=0,\dots ,k-1}
|A_{j}(te^{i\theta })|^{1/(k-j)}dt\Big),  \label{e2.10}
\end{equation}
where $C>0$ is a constant satisfying
\begin{equation}
C\leq (1+\varepsilon )\max_{j=0,\dots ,k-1}
\Big(\frac{|f^{(j)}(z_{\theta })|}{
(n_{c})^{j}\max\limits_{n=0,\dots ,k-1} |A_n(z_{\theta })|
^{j/(k-n)}}\Big).  \label{e2.11}
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2.5}
Let $p\geq  q\geq  1$ be integers. If
$A_0(z),\dots ,A_{k-1}(z)$ are analytic functions of
$[p,q]$-order in the unit disc $\Delta $, then every solution
$f\not\equiv 0$ of \eqref{e1.1} satisfies
\begin{equation}
\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q]
}(f)\leq \max \{\rho_{M,[p,q] }(
A_{j}):j=0,1,\dots ,k-1\}.  \label{e2.12}
\end{equation}
\end{lemma}

\begin{proof} Set $\sigma=\max \{\rho_{M,[p,q]}(A_{j}):
j=0,1,\dots ,k-1\}$. Let $f\not\equiv 0$
be a solution of \eqref{e1.1}. Let
$\theta_0\in [0,2\pi )$ be such that
$|f(re^{i\theta_0})|=M(r,f)$. By Lemma \ref{lem2.4}, we have
\begin{equation}
\begin{split}
M(r,f)&\leq C\exp \Big(n_{c} \int_\nu ^r
\max_{j=0,\dots ,k-1} |A_{j}(te^{i\theta})|^{1/(k-j)}dt\Big)\\
&\leq C\exp \big(n_{c} \int_\nu ^r
\max_{j=0,\dots ,k-1} (M(r,A_{j}))^{1/(k-j)}dt\big)\\
&\leq C\exp (n_{c}(r-\nu )\max_{j=0,\dots ,k-1}\{M(r,A_{j})\}).
\end{split} \label{e2.13}
\end{equation}
By Definition \ref{def1.4},
\begin{equation}
M(r,A_{j})\leq \exp_{p+1}\{(\sigma
+\varepsilon )\log_q(\frac{1}{1-r})\}\quad
(j=0,\dots ,k-1)\label{e2.14}
\end{equation}
holds for any $\varepsilon >0$. Hence from \eqref{e2.13} and
\eqref{e2.14} we obtain
\begin{equation}
\rho_{M,[p+1,q] }(f)\leq \sigma +\varepsilon .
\label{e2.15}
\end{equation}
Since $\varepsilon >0$ is arbitrary, we have by
 Proposition \ref{prop1.1} (ii)
\begin{equation*}
\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q]
}(f)\leq \sigma =\max \{\rho_{M,[p,q]
}(A_{j}):j=0,1,\dots ,k-1\}.
\end{equation*}
\end{proof}

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

 Suppose that $f\not\equiv 0$ is a solution of \eqref{e1.1}. By
\eqref{e1.1}, we can write
\begin{equation}
A_0(z)=-\Big(\frac{f^{(k)}}{f}+A_{k-1}(
z)\frac{f^{(k-1)}}{f}+\dots +A_1(z)\frac{
f'}{f}\Big).  \label{e3.1}
\end{equation}
From the condition \eqref{e1.4},  by using \eqref{e3.1}
and Lemma \ref{lem2.1} we obtain
\begin{equation}
\begin{split}
m(r,A_0)&\leq \sum_{j=1}^{k-1}m(r,A_{j})
+\sum_{j=1}^k  m\big(r,\frac{f^{(j)}}{f}\big)+O(1)\\
&\leq (k-1)\exp_p\{\beta \log_q(\frac{1}{1-r})\}+S(r,f)
\end{split}\label{e3.2}
\end{equation}
holds for all $r$ outside a set $E_1\subset [0,1)$ with
$\int_{E_1}\frac{dr}{1-r}<\infty $. By Lemma \ref{lem2.3}
 and \eqref{e3.2},
we have
\begin{equation}
\begin{split}
m(r,A_0)
&\leq (k-1)\exp_p\{\beta\log_q(\frac{1}{1-s(r)})\} \\
&\quad +O\Big(\log ^{+}T(s(r),f)+\log (\frac{1}{1-s(r)})\Big)
\end{split}\label{e3.3}
\end{equation}
holds for all $r\in [0,1)$. The assumption \eqref{e1.3} gives us
\begin{equation}
\begin{split}
m(r_n,A_0)&=T(r_n,A_0)
\geq  \exp_p\{\alpha \log_q\big(\frac{d}{1-s(r_n)}\big)\}\\
&\geq  \exp_p\{\gamma \log_q\big(\frac{1}{1-s(r_n)}\big)\},
\end{split}  \label{e3.4}
\end{equation}
where $\gamma $ is an arbitrary number satisfying
$\beta <\gamma <\alpha $ and $n$ is sufficiently large.
By combining \eqref{e3.3} and \eqref{e3.4} for $r=r_n$,
for $n$ sufficiently large we obtain
\begin{equation}
\begin{split}
\exp_p\{\gamma \log_q\big(\frac{1}{1-s(r_n)}\big)\}
&\leq (k-1)\exp_p\{\beta \log_q\big(\frac{1}{1-s(r_n)}\big)\}\\
&\quad +O(\log ^{+}T(s(r_n),f)+\log (\frac{1}{1-s(r_n)})).
\end{split} \label{e3.5}
\end{equation}
Noting that $\gamma >\beta \geq  0$, it follows from \eqref{e3.5}
that
\begin{equation}
(1-o(1))\exp_p\{\gamma \log_q\big( \frac{1}{1-s(r_n)}\big)\}
\leq O\Big(\log ^{+}T(s(r_n),f)+\log (\frac{1}{
1-s(r_n)})\Big)\label{e3.6}
\end{equation}
holds as $r_n\to 1^{-}$. Hence, by \eqref{e3.6} we obtain
$\rho_{[p,q]}(f)=\rho_{M,[p,q]}(f)=\infty $ and
\begin{equation*}
\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q]}(f)
=\limsup_{s(r_n)\to 1^{-}}
 \frac{\log_{p+1}^{+}T(s(r_n), f)}{\log_q\frac{1}{1-s(r_n)}}
\geq  \gamma .
\end{equation*}
Since $\gamma $ is an arbitrary number less than $\alpha $,
we obtain
$\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q] }(f)\geq  \alpha $.

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

 Suppose that $f\not\equiv 0$ is a solution of \eqref{e1.1}.
Then for any given $\varepsilon >0$, by the results of
Theorem \ref{thm1.1}, we have
$\rho_{[p,q] }(f)=\rho_{M,[p,q] }(f)=\infty $
and
\begin{equation}
\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q]
}(f)\geq  \rho -\varepsilon .  \label{e4.1}
\end{equation}
Since $\varepsilon >0$ is arbitrary, from \eqref{e4.1} we obtain
$\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q]}(f)\geq  \rho =\rho_{[p,q] }(
A_0)$. On the other hand, by Lemma \ref{lem2.5}, we have
\begin{equation}
\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q]
}(f)\leq \max \{\rho_{M,[p,q] }(
A_{j}):j=0,1,\dots ,k-1\}.  \label{e4.2}
\end{equation}
It yields
\begin{equation*}
\rho_{[p,q] }(A_0)\leq \rho_{[p+1,q
] }(f)=\rho_{M,[p+1,q] }(f)
\leq \max \{\rho_{M,[p,q] }(A_{j})
:j=0,1,\dots ,k-1\}.
\end{equation*}
If $p>q$, then
\begin{equation*}
\max \{\rho_{M,[p,q] }(A_{j})
:j=0,1,\dots ,k-1\}=\rho_{[p,q] }(A_0).
\end{equation*}
Therefore,
\begin{equation*}
\rho_{[p+1,q] }(f)=\rho_{M,[p+1,q]
}(f)=\rho_{[p,q] }(A_0).
\end{equation*}

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

 Set $\max \{\rho_{[p,q] }(A_{j})
(j\neq s),\rho_{[p,q] }(F)\}
=\beta <\rho_{[p,q] }(A_{s})=\alpha $. Suppose
that $f$ is an admissible solution of \eqref{e1.2} with
$\rho =\rho_{[p,q] }(f)<\infty $. It follows from \eqref{e1.2}
that
\begin{equation}
\begin{split}
A_{s}(z)
&=\frac{F(z)}{f^{(s)}}-\frac{ f^{(k)}}{f^{(s)}}-A_{k-1}(z)\frac{
f^{(k-1)}}{f^{(s)}}-\dots -A_{s+1}(z)
\frac{f^{(s+1)}}{f^{(s)}}\\
&\quad -A_{s-1}(z)\frac{f^{(s-1)}}{f^{(s)}}
-\dots -A_1(z)\frac{f'}{f^{(s)}}
-A_0(z)\frac{f}{f^{(s)}}.
\end{split} \label{e5.1}
\end{equation}
Applying Lemma \ref{lem2.2}, we have
\begin{equation}
m\big(r,\frac{f^{(j+1)}}{f}\big)
=O\Big(\exp_{p-1}\{(\rho +\varepsilon )
\log_q(\frac{1}{1-r} )\}\Big)\quad
 (j=0,\dots ,k-1) \label{e5.2}
\end{equation}
holds for all $r$ outside a set $E_2\subset [0,1)$ with
 $\int_{E_2}\frac{dr}{1-r}<\infty $. Since $N(r,f^{(j+1)
})=0$, it holds for $j=0,\dots ,k-1$ that
\begin{equation}
\begin{split}
T(r,f^{(j+1)})
&=m(r,f^{(j+1) })\leq m\big(r,\frac{f^{(j+1)}}{f}\big) +m(r,f)\\
&\leq T(r,f)+m\big(r,\frac{f^{(j+1)}}{f}\big).
\end{split} \label{e5.3}
\end{equation}
By \eqref{e5.3}, from \eqref{e5.1} and \eqref{e5.2} we obtain
\begin{equation}
\begin{split}
T(r,A_{s})
&\leq T(r,F)+cT(r,f)+ \sum_{j\neq s} T(r,A_{j}) \\
&\quad +O\Big(\exp_{p-1}\{(\rho +\varepsilon )\log_q(
\frac{1}{1-r})\}\Big)\quad (r\notin E_2),
\end{split}\label{e5.4}
\end{equation}
where $c>0$ is a constant. Since
$\rho_{[p,q] }(A_{s})=\alpha $, there exists a sequence
$\{r_n'\}$ $(r_n'\to 1^{-})$ such that
\begin{equation}
\lim_{r_n'\mapsto 1^{-}} \frac{\log_p^{+}T(
r_n',A_{s})}{\log_q\frac{1}{1-r_n'}}=\alpha .
\label{e5.5}
\end{equation}
Set $\int_{E_2}\frac{dr}{1-r}:=\log \gamma <\infty $. Since
$\int_{r_n'}^{1-\frac{1-r_n'}{\gamma +1}}\frac{dr
}{1-r}=\log (\gamma +1)$,  there exists a point $r_n\in
[r_n',1-\frac{1-r_n'}{\gamma +1}]
-E_2\subset [0,1)$. From
\begin{equation}
\frac{\log_p^{+}T(r_n,A_{s})}{\log_q\frac{1}{1-r_n}}
\geq  \frac{\log_p^{+}T(r_n',A_{s})}{\log
 _q(\frac{\gamma +1}{1-r_n'})}
=\frac{\log _p^{+}T(r_n',A_{s})}{\log_q\frac{1}{
1-r_n'}+\log \big(\frac{\log_{q-1}(\frac{\gamma +1}{
1-r_n'})}{\log_{q-1}\frac{1}{1-r_n'}}
\big)},  \label{e5.6}
\end{equation}
it follows that
\begin{equation}
\lim_{r_n\to 1^{-}} \frac{\log_p^{+}T(r_n,A_{s})}{\log_q\frac{1}{1-r_n}}
=\alpha .  \label{e5.7}
\end{equation}
So, for any given $\varepsilon$ $(0<2\varepsilon <\alpha -\beta )$,
we have
\begin{equation}
T(r_n,A_{s})>\exp_p\{(\alpha -\varepsilon
)\log_q(\frac{1}{1-r_n})\}\label{e5.8}
\end{equation}
and for $j\neq s$,
\begin{gather}
T(r_n,A_{j})\leq \exp_p\{(\beta
+\varepsilon )\log_q(\frac{1}{1-r_n})\},  \label{e5.9}\\
T(r_n,F)\leq \exp_p\{(\beta +\varepsilon
)\log_q(\frac{1}{1-r_n})\}\label{e5.10}
\end{gather}
hold as $r_n\to 1^{-}$. By \eqref{e5.4}, \eqref{e5.8},
 \eqref{e5.9} and \eqref{e5.10},  for
$r_n\to 1^{-}$ we can obtain
\begin{equation}
\begin{split}
\exp_p\{(\alpha -\varepsilon )\log_q(\frac{1}{1-r_n})\}
&\leq k\exp_p\{(\beta+\varepsilon )\log_q(\frac{1}{1-r_n})\}
+cT(r_n,f)\\
&\quad +O\Big(\exp_{p-1}\{(\rho +\varepsilon )
\log_q(\frac{1}{1-r_n})\}\Big)).
\end{split} \label{e5.11}
\end{equation}
Noting that $\alpha -\varepsilon >\beta +\varepsilon $,
it follows from \eqref{e5.11} that for $r_n\to 1^{-}$,
\begin{equation}
\begin{split}
&(1-o(1))\exp_p\{(\alpha-\varepsilon )\log_q(\frac{1}{1-r_n})\}\\
&\leq cT(r_n,f)
+O\Big(\exp_{p-1}\{(\rho +\varepsilon )\log_q(
\frac{1}{1-r_n})\}\Big).
\end{split} \label{e5.12}
\end{equation}
Therefore, by \eqref{e5.12} we obtain
\begin{equation*}
\limsup_{r_n\mapsto 1^{-}} \frac{\log_p^{+}T(
r_n,f)}{\log_q\frac{1}{1-r_n}}\geq  \alpha -\varepsilon
\end{equation*}
and since $\varepsilon >0$ is arbitrary, we obtain
$\rho_{[p,q]}(f)\geq  \rho_{[p,q] }(A_{s})=\alpha $. This
completes the proof.

\section{Proof of Theorem \ref{thm1.4}}

 Setting $\max \{\rho_{[p,q] }(A_{j})(j\neq s),\rho_{[p,q] }(
F)\}=\beta $,  for a given $\varepsilon >0$, we have
\begin{gather}
T(r,A_{j})\leq \exp_p\{(\beta +\varepsilon )\log
_q(\frac{1}{1-r})\}\text{ }(j\neq s), \label{e6.1}\\
T(r,F)\leq \exp_p\{(\beta +\varepsilon )\log
_q(\frac{1}{1-r})\}\label{e6.2}
\end{gather}
as $r\to 1^{-}$. Now  from \eqref{e1.2} we can write
\begin{equation}
\begin{split}
A_{s}(z)&=\frac{F(z)}{f^{(s)}}-\frac{
f^{(k)}}{f^{(s)}}-A_{k-1}(z)\frac{
f^{(k-1)}}{f^{(s)}}-\dots -A_{s+1}(z)
\frac{f^{(s+1)}}{f^{(s)}} \\
&\quad -A_{s-1}(z)\frac{f^{(s-1)}}{f^{(s)}}
-\dots -A_1(z)\frac{f'}{f^{(s)}}
-A_0(z)\frac{f}{f^{(s)}}.
\end{split}  \label{e6.3}
\end{equation}
Hence by \eqref{e5.3} and \eqref{e6.3} we obtain
\begin{equation}
T(r,A_{s})\leq T(r,F)+cT(r,f)+
\sum_{j=0}^{k-1}  m\big(r,\frac{f^{(j+1)}}{f}\big)
+\sum_{j\neq s} T(r,A_{j}),  \label{e6.4}
\end{equation}
where $c>0$ is a constant. If $\rho =\rho_{[p,q] }(f)<\infty $,
then by Lemma \ref{lem2.2}
\begin{equation}
m\big(r,\frac{f^{(j+1)}}{f}\big)=O\Big(\exp
_{p-1}\{(\rho +\varepsilon )\log_q(\frac{1}{1-r}
)\}\Big)\quad (j=0,\dots ,k-1)\label{e6.5}
\end{equation}
holds for all $r$ outside a set $E_2\subset [0,1)$ with
$\int_{E_2}\frac{dr}{1-r}<\infty $. For $r\to 1^{-}$, we have
\begin{equation}
T(r,f)\leq \exp_p\{(\rho +\varepsilon
)\log_q(\frac{1}{1-r})\}.  \label{e6.6}
\end{equation}
Thus, by \eqref{e6.1}, \eqref{e6.2}, \eqref{e6.4}, \eqref{e6.5}
 and \eqref{e6.6}, we obtain
\begin{equation}
\begin{split}
T(r,A_{s})
&\leq k\exp_p\{(\beta +\varepsilon
)\log_q(\frac{1}{1-r})\}+c\exp_p\{
(\rho +\varepsilon )\log_q(\frac{1}{1-r})\} \\
&\quad +O(\exp_{p-1}\{(\rho +\varepsilon )\log_q(
\frac{1}{1-r})\})
\end{split}\label{e6.7}
\end{equation}
for $r\notin E_2$ and $r\to 1^{-}$. By Lemma \ref{lem2.3},  for any
$d\in [0,1)$, we have
\begin{equation}
\begin{split}
T(r,A_{s})
&\leq k\exp_p\{(\beta +\varepsilon)\log_q(\frac{1}{d(1-r)})\}\\
&\quad +c\exp_p\{(\rho +\varepsilon )\log_q\big(\frac{1}{
d(1-r)}\big)\} \\
&\quad
+O\Big(\exp_{p-1}\{(\rho +\varepsilon )\log_q(
\frac{1}{d(1-r)})\}\Big)
\end{split}\label{e6.8}
\end{equation}
as $r\to 1^{-}$. Therefore,
\begin{equation*}
\rho_{[p,q] }(A_{s})\leq \max \{\beta
+\varepsilon ,\rho +\varepsilon \}<\infty .
\end{equation*}
This contradicts  that $\rho_{[p,q] }(A_{s})=\infty $.
This completes the proof



\subsection*{Acknowledgments}
The author wants to thank the anonymous referee for his/her
valuable suggestions and helpful remarks. This research is
supported by ANDRU (Agence Nationale pour le D\'{e}veloppement de la
Recherche Universitaire) and University of Mostaganem (UMAB),
(PNR Project Code 8/u27/3144).

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