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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 65, 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 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/65\hfil Growth of solutions]
{Growth of solutions of higher-order linear
differential equations}

\author[K. Hamani\hfil EJDE-2010/65\hfilneg]
{Karima Hamani} 

\address{Karima Hamani \newline
Department of Mathematics\\
Laboratory of Pure and Applied Mathematics\\
University of Mostaganem, B. P. 227 Mostaganem, Algeria}
\email{hamanikarima@yahoo.fr}

\thanks{Submitted February 3, 2010. Published May 8, 2010.}
\subjclass[2000]{34A20, 30D35}
\keywords{Linear differential equation; entire function;
hyper-order}

\begin{abstract}
 In this article, we study the growth of
 solutions of the linear differential equation
 \[
 f^{(k)}+(A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}(z)) f^{(k-1)}+\dots
 +(A_0(z)e^{P_0(z)}+B_0(z))f=0,
 \]
 where $k\geq 2$ is an integer, $P_j(z)$  are nonconstant
 polynomials  and  $A_j(z), B_j(z)$ are entire functions,
 not identically zero. We determine the hyper-order of these
 solutions, under certain conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction and statement of results}


In this article, we assume that the reader is familiar with the
fundamental results and  standard notation of the Nevanlinna value
distribution theory of meromorphic functions  \cite{h1}. Let
$\sigma (f)$ denote the order of growth of an entire function
$f(z)$ and $\sigma _2(f)$ the hyper-order of $f(z)$, which
as in \cite{k1,y1} is defined by
\begin{equation}
\sigma _2(f)=\limsup_{r\to +\infty}\frac{\log \log T(r,f)}{\log
r}=\limsup_{r\to +\infty}\frac{\log \log \log M(r,f)}{\log r},
\label{e1.1}
\end{equation}
where $M(r,f)=\max_{|z|=r} | f( z)|$.

We define the linear measure of a set $E\subset [ 0,+\infty )$ by
$m(E)=\int_0^{+\infty }\chi _E(t)dt$ and the logarithmic measure
of a set $H\subset [ 1,+\infty )$ by $lm(H)=\int_1^{+\infty
}\frac{\chi _{H}(t)}{t}dt$, where $\chi _E$ is the characteristic
function of a set $E$ .

Several authors \cite{c1,g2,k1}  have studied the second-order linear
differential equation
\begin{equation}
f''+h_1(z)e^{P(z)}f'+h_0(z)e^{Q(z)}f=0, \label{e1.2}
\end{equation}
where $P(z)$ and $Q(z)$ are nonconstant polynomials, $h_1(z)$ and
$h_0(z)\not\equiv 0$ are entire functions satisfying $\sigma
(h_1)<\deg P$ and $\sigma (h_0)<\deg Q$. Gundersen showed in
\cite[p. 419]{g2}  that if $\deg P\neq \deg Q$, then every nonconstant
solution of  \eqref{e1.2} is of
infinite order. If $\deg P=\deg Q$, then  \eqref{e1.2} can
have nonconstant solutions of finite order. Indeed, $f(z)=z$
satisfies $f''-z^{3}e^{z}f'+z^{2}e^{z}f=0$.
Kwon \cite{k1} studied the case where $\deg P=\deg Q$ and proved
the following result:

\begin{theorem}[\cite{k1}] \label{thmA}
Let $P(z)$ and $Q(z)$ be nonconstant polynomials such that
\begin{gather}
P(z)=a_{n}z^{n}+\dots +a_1z+a_0,  \label{e1.3} \\
Q(z)=b_{n}z^{n}+\dots +b_1z+b_0,  \label{e1.4}
\end{gather}
where $a_{i},b_{i}$ $(i=0,1,\dots ,n)$ are
complex numbers, $a_{n}\neq 0$ and $b_{n}\neq 0$.
Let $h_j(z)$  $(j=0,1)$ be entire
functions with $\sigma ( h_j)<n$. Suppose that
$\arg a_{n}\neq \arg b_{n}$ or $a_{n}=cb_{n}$ $(0<c<1)$.
Then every nonconstant solution $f$ of
\eqref{e1.2} is of infinite order and satisfies
$\sigma _2(f)\geq n$.
\end{theorem}

Chen \cite{c2} also studied the growth of solutions of  second-order
linear differential equations and obtained the following result:

\begin{theorem}[\cite{c2}] \label{thmB}
Let  $A_j(z)$
$(\not\equiv 0)$, $D_j(z) $ $(j=0,1)$ be entire
functions with $\sigma (A_j)<1$, $\sigma ( D_j)
<1$, $a$, $b$ be complex constants such
that  $ab\neq 0$  and $\arg a\neq \arg b$ or
$a=cb$  $(0<c<1)$.  Then every solution $f$
$(\not\equiv 0)$ of the equation
\begin{equation}
f''+(A_1(z)e^{az}+D_1(z))f'+(A_0(z)e^{bz}+D_0(z))f=0 \label{e1.5}
\end{equation}
is of infinite order.
\end{theorem}

Belaidi \cite{b1} extended Theorem \ref{thmB} for higher-order linear differential
equations as follows.

\begin{theorem}[\cite{b1}] \label{thmC}
 Let  $k\geq 2$ and
$P_j(z)=\sum_{i=0}^n a_{i,j}z^{i}$
$(j=0,1,\dots ,k-1)$ be nonconstant polynomials, where
$a_{0,j},\dots ,a_{n,j}$ $( j=0,\dots ,k-1)$  are complex
numbers such that $a_{n,j}\neq 0$  $(j=0,1,\dots ,k-1)$.
Let $A_j(z)$ $(\not\equiv 0)$, $B_j(z)$ $(\not\equiv
0)$  $(j=0,1,\dots ,k-1)$ be entire functions. Suppose
that $\arg a_{n,j}\neq \arg a_{n,0}$ or
$a_{n,j}=c_ja_{n,0}$  $(0<c_j<1)$ $(j=1,\dots ,k-1)$
and $\sigma (A_j)<n$, $\sigma (B_j)<n$
$(j=0,1,\dots ,k-1)$.  Then every solution $f$
$(\not\equiv 0)$  of the differential equation
\begin{equation}
f^{(k)}+(A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}(z)) f^{(k-1)}+\dots
+(A_0(z)e^{P_0(z)}+B_0(z))f=0 \label{e1.6}
\end{equation}
is of infinite order and satisfies $\sigma _2(f)=n$.
\end{theorem}

Chen \cite{c3} also considered the growth of solutions of
 higher-order linear differential equations and proved the
following result:

\begin{theorem}[\cite{c3}] \label{thmD}
Let  $h_j(z)$
$(j=0,1,\dots ,k-1)$ $(k\geq 2) $  be entire functions
with $\sigma (h_j)<1$,  and
$H_j(z)=h_j(z)e^{a_jz}$, where $a_j$
$(j=0,1,\dots ,k-1)$ are complex numbers. Suppose that
there exists $a_s$ such that $h_s\not\equiv 0$,
 for $j\neq s$, if $H_j\not\equiv 0$,
$a_j=c_ja_s $ $(0<c_j<1)$; if
$H_j\equiv 0$, we define  $c_j=0$. Then
every transcendental solution $f$ of the equation
\begin{equation}
f^{(k)}+H_{k-1}(z)f^{(k-1)}+\dots +H_s(z)f^{( s) }+\dots
+H_0(z)f=0 \label{e1.7}
\end{equation}
is of infinite order.

Furthermore, if $\max \{ c_1,\dots c_{s-1}\}<c_0$, then every solution
of  \eqref{e1.7} is of infinite order.
\end{theorem}


Recently,  Tu and  Yi \cite{t1} obtained the following result
which is an extension of Theorem \ref{thmA}.


\begin{theorem}[\cite{t1}] \label{thmE} Let $h_j(z)$
$(j=0,1,\dots ,k-1)$ $( k\geq 2)$ be entire functions with
$\sigma (h_j)<n$ $(n\geq 1) $, and let
$P_j(z)=\sum_{i=0}^n a_{i,j}z^{i} $
$(j=0,1,\dots ,k-1)$ be polynomials with degree
$n$ , where $a_{n,j}$ $(j=0,1,\dots ,k-1)$ are
complex numbers such that $a_{n,0}=|a_{n,0}|e^{i\theta
_0}$, $a_{n,s}=|a_{n,s}|e^{i\theta _s}$,
$a_{n,0}a_{n,s}\neq 0$ $(0<s\leq k-1)$,  $\theta
_0$, $\theta _s\in [ 0,2\pi )$,  $\theta
_0\neq \theta _s$, $h_0h_s\not\equiv 0$; for
$j\neq 0,s$, $a_{n,j}$ satisfies either
$a_{n,j}=c_ja_{n,0}$ $(c_j<1)$ or
$\arg a_{n,j}=\theta _s$. Then every solution $f$
$(\not\equiv 0)$ of the equation
\begin{equation}
f^{(k)}+h_{k-1}(z)e^{P_{k-1}(z)}f^{(k-1)}+\dots
+h_1(z)e^{P_1(z)}f'+h_0(z)e^{P_0(z)}f=0 \label{e1.8}
\end{equation}
is of infinite order and satisfies $\sigma _2(f)=n$.
\end{theorem}

The main purpose of this article is to investigate the growth of
solutions of  \eqref{e1.6}, and determine the hyper-order
of these solutions. We shall prove the following results:


\begin{theorem} \label{thm1.1}
Let $k\geq 2$ be an integer and $P_j(z)=\sum_{i=0}^n a_{i,j}z^{i}$
$(j=0,1,\dots ,k-1)$ be nonconstant
polynomials, where  $a_{0,j},a_{1,j},\dots ,a_{n,j}$
$(j=0,1,\dots ,k-1)$ are complex numbers such that
$a_{n,j}\neq 0$ $(j=0,1,\dots ,k-1) $. Let $A_j(z)$
$(\not\equiv 0)$, $B_j(z)$  $(\not\equiv 0)$ $(j=0,1,\dots
,k-1)$ be entire functions with $\sigma (A_j)<n$ and
$\sigma (B_j)<n$. Suppose that
there exists $s\in \{ 1,\dots ,k-1\} $ such
that $\arg a_{n,j}\neq \arg a_{n,s}$ $(j\neq s)$. Then
every transcendental solution $f$ of
\eqref{e1.6} is of infinite order and satisfies
$\sigma_2(f)=n$.
\end{theorem}

\begin{theorem} \label{thm1.2}
Let $k\geq 2$ be an integer and $P_j(z)=\sum_{i=0}^n a_{i,j}z^{i}$
$(j=0,1,\dots ,k-1)$ be nonconstant
polynomials, where  $a_{0,j},a_{1,j},\dots ,a_{n,j}$
$(j=0,1,\dots ,k-1)$ are complex numbers such that
$a_{n,j}\neq 0$ $(j=0,1,\dots ,k-1)$. Let $A_j(z)$
$(\not\equiv 0)$, $B_j(z)$  $(\not\equiv 0)$ $(j=0,1,\dots
,k-1)$ be entire functions with $\sigma (A_j)<n$ and
$\sigma (B_j)<n$. Suppose that
there exists $s\in \{ 1,\dots ,k-1\} $ such
that $a_{n,j}=c_ja_{n,s}$ $(0<c_j<1)$ $(j\neq s)$.
Then every transcendental solution $f$ of
\eqref{e1.6} is of infinite order and satisfies
 $\sigma _2(f)=n$.

Furthermore, if $\max \{ c_1,\dots c_{s-1}\}<c_0$, then every
solution of  \eqref{e1.6}
 is of infinite order and satisfies $\sigma _2(f)=n$.
\end{theorem}

\begin{theorem} \label{thm1.3}
Let $k\geq 2$ be an integer and
$P_j(z)=\sum_{i=0}^n a_{i,j}z^{i}$  $(j=0,1,\dots ,k-1)$
 be nonconstant polynomials, where  $a_{0,j},a_{1,j},\dots ,a_{n,j}$
$(j=0,1,\dots ,k-1)$ are complex numbers such that
$a_{n,j}\neq 0$ $(j=0,1,\dots ,k-1)$  and
$a_{n,0}=|a_{n,0}|e^{i\theta _0}$, $\theta _0\in [ 0,2\pi )$.
Let $A_j(z)$  $(\not\equiv 0)$, $B_j(z)$ $(\not\equiv 0) $
$(j=0,1,\dots ,k-1)$ be
entire functions with $\sigma (A_j)<n$ and
$\sigma (B_j)<n$ $(j=0,1,\dots ,k-1)$. Suppose that there
exists $s\in \{ 1,\dots ,k-1\} $  such that
$a_{n,s}=|a_{n,s}|e^{i\theta _s} $,
$\theta _s\in [0,2\pi )$, $\theta _s\neq \theta _0$ and for
$j\in \{ 1,\dots ,s-1,s+1,\dots ,k-1\} $,
$a_{n,j}$ satisfies either $a_{n,j}=c_ja_{n,0}$
$(c_j<1)$ or $\arg a_{n,j}=\theta _s$. Then
every solution $f$ $(\not\equiv 0)$ of
\eqref{e1.6}  is of infinite order and satisfies
$\sigma _2(f)=n$.
\end{theorem}

\begin{theorem} \label{thm1.4}
Let $k\geq 2$ be an integer and $P_j(z)=\sum_{i=0}^n a_{i,j}z^{i}$
$(j=0,1,\dots ,k-1)$ be nonconstant
polynomials, where  $a_{0,j},a_{1,j},\dots ,a_{n,j}$
$(j=0,1,\dots ,k-1)$ are complex numbers such that
$a_{n,j}\neq 0$ $(j=0,1,\dots ,k-1)$ . Let $A_j(z)$
$(\not\equiv 0)$, $B_j(z)$  $(\not\equiv 0)$
$(j=0,1,\dots ,k-1)$ be entire functions with
$\sigma (A_j)<n$\ and $\sigma (B_j)<n$
$(j=0,1,\dots ,k-1)$. Suppose that there exist $d$,
$s\in \{ 1,\dots ,k-1\} $ such that
$a_{n,d}=|a_{n,d}|e^{i\theta _d}$,
$a_{n,s}=|a_{n,s}|e^{i\theta _s}$, $\theta _d$,
$\theta _s\in [ 0,2\pi )$, $\theta _d\neq \theta _s$
 and for $j\in \{ 0,\dots ,k-1\} \diagdown
\{ d,s\} $, $a_{n,j}$ satisfies either
$a_{n,j}=c_ja_{n,d}$ $(c_j<1)$ or
$\arg a_{n,j}=\theta _s$. Then every transcendental solution
$f$ of  \eqref{e1.6} is of infinite
order and satisfies $\sigma _2(f)=n$.
\end{theorem}

\section{Preliminaries}

 \begin{lemma}[\cite{g1}] \label{lem2.1}
Let $f(z)$ be a transcendental meromorphic function and let
$\alpha >1$ and $\epsilon >0$\ be given constants.
Then there exist a set  $E_1\subset [ 1,+\infty )$
having finite logarithmic measure and a constant
$B>0$ that depends only on $\alpha $ and
$(i,j)$ ($i,j$ positive integers with
$i>j$) such that for all $z$\ satisfying
$|z|=r\notin [ 0,1] \cup E_1$, we have
\begin{equation}
\big|\frac{f^{(i)}(z)}{f^{(j)}(z)} \big|
\leq B\big[ \frac{T(\alpha
r,f)}{r}(\log ^{\alpha }r)\log T(\alpha r,f)\big] ^{i-j}.
\label{e2.1}
\end{equation}
\end{lemma}

 \begin{lemma}[\cite{c3}] \label{lem2.2}
Let $f(z)$ be a transcendental entire function Then there
exists a set $E_2\subset [ 1,+\infty )$ that has
finite logarithmic measure such that for all $z$
satisfying  $|z|=r\notin [ 0,1] \cup E_2$ and
$|f(z)|=M(r,f)$, we have
\begin{equation}
\big|\frac{f(z)}{f^{(s)}(z)} \big|
\leq 2r^{s} ,\label{e2.2}
\end{equation}
where $s\geq 1$ is an integer.
\end{lemma}

 \begin{lemma}[\cite{m1}] \label{lem2.3}
Let  $P(z)=(\alpha +i\beta )z^{n}+\dots $
($\alpha $, $\beta $ are real numbers,
$|\alpha |+|\beta |\neq 0$) be a polynomial with degree
$n\geq 1$  and $A(z)$ be an entire function
with $\sigma (A)<n$. Set $g(z)=A(z)e^{P( z)}$,
$z=re^{i\theta }$, $\delta ( P,\theta )=\alpha \cos n\theta -\beta
\sin n\theta $. Then for any given $\epsilon >0$,
 there exists a set  $E_3\subset [ 1,+\infty)$
having finite logarithmic measure such that for any
$\theta \in [ 0,2\pi )\setminus H$  $(H=\{ \theta \in [
0,2\pi ):\delta ( P,\theta ) =0\} )$ and for
$|z|=r\notin [ 0,1] \cup E_3$, we have
\begin{itemize}
\item[(i)] if $\delta (P,\theta )>0$, then
\begin{equation}
\exp \{ (1-\epsilon )\delta (P,\theta ) r^{n}\} \leq
|g(re^{i\theta })|\leq \exp \{ (1+\epsilon )\delta
(P,\theta )r^{n}\} , \label{e2.3}
\end{equation}

\item[(ii)] if $\delta (P,\theta )<0$, then
\begin{equation}
\exp \{ (1+\epsilon )\delta (P,\theta ) r^{n}\} \leq
|g(re^{i\theta })|\leq \exp \{ (1-\epsilon )\delta
(P,\theta )r^{n}\} . \label{e2.4}
\end{equation}
\end{itemize}
\end{lemma}

 \begin{lemma}[\cite{c3}] \label{lem2.4}
 Let  $k\geq 2$ be an integer and let $A_j(z)$ $(j=0,1,\dots
,k-1)$ be entire functions of finite order. If
$f$  is a solution of the differential equation
\begin{equation}
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots +A_1( z)f'+A_0(z)f=0,
\label{e2.5}
\end{equation}
then $\sigma _2(f)\leq \max \{\sigma (A_j)$
$(j=0,1,\dots ,k-1)\}$.
\end{lemma}

\section{Proof of main results}

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

Assume $f$ is a transcendental solution of  \eqref{e1.6}.
By Lemma \ref{lem2.1}, there exist a constant $B>0$ and a set
$E_1\subset [ 1,+\infty )$ having finite logarithmic measure such
that for all $z$ satisfying $|z|=r\notin [ 0,1]\cup E_1$, we have
\begin{gather}
\big|\frac{f^{(j)}(z)}{f^{(s)}(z)}\big|\leq Br [ T(2r,f)]
^{j-s+1}\quad (j=s+1,\dots,k),  \label{e3.1}
\\
|\frac{f^{(j)}(z)}{f(z)}|\leq Br[ T(2r,f)] ^{j+1}\quad (j=1,2,\dots,s-1).
\label{e3.2}
\end{gather}
By Lemma \ref{lem2.2}, there exists a set $E_2\subset [ 1,+\infty )$
that has finite logarithmic measure such that for all $z$
satisfying $|z|=r\notin [ 0,1] \cup E_2$ and $|f(z)| =M(r,f)$,
we have
\begin{equation}
\big|\frac{f(z)}{f^{(s)}(z)} \big|\leq 2r^{s}. \label{e3.3}
\end{equation}
Since $\arg a_{n,j}\neq \arg a_{n,s}$ $(j\neq s)$, there is a ray
$\arg z=\theta \in [ 0,2\pi )\setminus H$, where
$H=\{\theta \in [ 0,2\pi ):\delta (P_0,\theta )=0$ or $\dots$ or
$\delta (P_{k-1},\theta )=0$\}, such that $\delta ( P_s,\theta )>0$,
$\delta (P_j,\theta )<0$ $(j\neq s)$. Set
$\beta =\max \{ \sigma ( B_j)\; (j=0,\dots ,k-1)\} $.
By Lemma \ref{lem2.3}, for any given $\epsilon $
($0<2\epsilon <\min \{ 1,n-\beta\} $), there exists a set
$E_3\subset [ 1,+\infty )$ having finite logarithmic measure
such that for all $z$ with $\arg z=\theta $,
$|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently
large $r$, we have
\begin{equation}
|A_s(z)e^{P_s(z)}+B_s(z)|\geq (1-o(1))\exp \{
(1-\epsilon )\delta (P_s,\theta ) r^{n}\} \label{e3.4}
\end{equation}
and
\begin{equation}
\begin{aligned}
|A_j(z)e^{P_j(z)}+B_j(z)|
&\leq  \exp \{ (1-\epsilon)\delta (P_j,\theta )r^{n}\}
+\exp \{ r^{\sigma (B_j)+\frac{\epsilon }{2}}\}   \\
&\leq \exp \{ r^{\sigma (B_j)+\epsilon }\}\\
&\leq \exp \{ r^{\beta +\epsilon }\} \quad (j\neq s).
\end{aligned}\label{e3.5}
\end{equation}
We can rewrite \eqref{e1.6} as
\begin{equation}
\begin{aligned}
&A_s(z)e^{P_s(z)}+B_s(z)\\
&=\frac{f^{(k)}}{f^{( s) }}+(A_{k-1}(z)e^{P_{k-1}(z)}
+B_{k-1}(z))\frac{f^{(k-1)}}{f^{(s)}} +\dots \\
&\quad+ (A_{s+1}(z)e^{P_{s+1}(z)}+B_{s+1}(z))
 \frac{ f^{(s+1)}}{f^{(s)}}+(A_{s-1}(z)e^{P_{s-1}(z)}
 +B_{s-1}(z))\frac{f^{(s-1)}}{f} \frac{f}{f^{(s)}}
\\
&\quad +\dots +(A_1(z)e^{P_1(z)}+B_1(z))\frac{ f'}{f}\frac{f}{f^{(s)}}
+(A_0(z)e^{P_0(z)}+B_0(z))\frac{f}{f^{( s)}}.
\end{aligned} \label{e3.6}
\end{equation}
Hence from \eqref{e3.1}-\eqref{e3.6}, for all $z$ with
$\arg z=\theta $, $|z|=r\notin [ 0,1] \cup E_1\cup E_2\cup
E_3$, $|f( z) |=M(r,f)$ and a sufficiently large $r$,
 we have
\begin{equation}
(1-o(1))\exp \{ (1-\epsilon )\delta (P_s,\theta )r^{n}\}
\leq M_1r^{s+1}\exp \{ r^{\beta +\epsilon
}\} [ T(2r,f)] ^{k}, \label{e3.7}
\end{equation}
where $M_1$ is some positive constant.
Thus $0<2\epsilon <\min \{ 1,n-\beta \} $ implies
$\sigma (f)=+\infty $ and $\sigma _2(f)\geq n$. By Lemma \ref{lem2.4},
we have $\sigma _2(f)=n$.

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

Assume $f$ is a transcendental solution of   \eqref{e1.6}.
Since $a_{n,j}=c_ja_{n,s}$ $(0<c_j<1)$ $(j\neq s)$, it
follows that $\delta (P_j,\theta )=c_j\delta (P_s,\theta )$
$(j\neq s)$. Put $c=\max \{ c_j (j\neq s)\} $. Then
$0<c<1.$We take a ray $\arg z=\theta \in [ 0,2\pi )\setminus H$,
where $H=\{ \theta \in [ 0,2\pi ) :\delta (P_s,\theta)=0\} $,
such that $\delta (P_s,\theta )>0$. By Lemma \ref{lem2.3},
for any given $\epsilon $ $(0<2\epsilon <\frac{1-c}{1+c})$,
there exists a set $E_3\subset [ 1,+\infty )$ having finite
logarithmic measure  such that for all $z$ with $\arg z=\theta $,
$|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we
have
\begin{equation}
|A_s(z)e^{P_s(z)}+B_s(z)|\geq (1-o(1))\exp \{
(1-\epsilon )\delta (P_s,\theta ) r^{n}\} \label{e4.1}
\end{equation}
and
\begin{equation}
|A_j(z)e^{P_j(z)}+B_j(z)|\leq (1+o(1))\exp \{
(1+\epsilon )c\delta (P_s,\theta ) r^{n}\} \quad (j\neq s).\label{e4.2}
\end{equation}
Thus by \eqref{e3.1}-\eqref{e3.3}, \eqref{e3.6}, \eqref{e4.1}
and \eqref{e4.2}, we obtain that for all $z$ with $\arg z=\theta $,
$|z|=r\notin [ 0,1] \cup E_1\cup E_2\cup E_3$, $|f(z)|=M(r,f)$
and a sufficiently large $r$,
\begin{equation}
\begin{aligned}
&(1-o(1))\exp \{ (1-\epsilon
)\delta (P_s,\theta )r^{n}\}   \\
&\leq  M_2r^{s+1}(1+o(1))\exp \{ ( 1+\epsilon ) c\delta
(P_s,\theta )r^{n}\} [ T(2r,f)] ^{k},
\end{aligned} \label{e4.3}
\end{equation}
where $M_2$  is a positive constant. By
$0<2\epsilon <\frac{1-c}{1+c}$ and \eqref{e4.3}, we have
\begin{equation}
\exp \{ \frac{(1-c)}{2}\delta (P_s,\theta ) r^{n}\}
\leq M_3r^{s+1}[ T(2r,f)] ^{k}, \label{e4.4}
\end{equation}
where $M_3$  is a positive constant.  Hence \eqref{e4.4}
 implies $\sigma (f)=+\infty $ and $\sigma _2( f)\geq n$.
By Lemma \ref{lem2.4}, we have $\sigma _2(f)=n$.


Now we prove that if $\max \{ c_1,\dots c_{s-1}\}
<c_0$, then equation \eqref{e1.6} cannot have a nonzero
polynomial solution. Suppose that $c'=\max \{ c_1,\dots
c_{s-1}\} <c_0$ and let $f(z)$ be a nonzero polynomial
solution of equation \eqref{e1.6} with $\deg f(z)=m$. We take a
ray $\arg z=\theta \in [ 0,2\pi )\setminus H$, where $H$ is
defined as above, such that $\delta (P_s,\theta )>0$.
By Lemma \ref{lem2.3}, for
any given $\epsilon $ $(0<2\epsilon <\min \{
\frac{1-c}{1+c} ,\frac{c_0-c'}{c_0+c'}\} )$, there exists a
set $E_3\subset [ 1,+\infty )$ having finite logarithmic measure
such that for all $z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1]
\cup E_3$ and a sufficiently large $r$, we have \eqref{e4.1}
and \eqref{e4.2}.


 If $m\geq s$, by \eqref{e1.6}, \eqref{e4.1} and
\eqref{e4.2}, we obtain for all $z$ with $\arg z=\theta $,
$|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$,
\begin{equation}
\begin{aligned}
&d_1r^{m-s}(1-o(1))\exp \{ ( 1-\epsilon )\delta
(P_s,\theta )r^{n}\}\\
&\leq |A_s(z)e^{P_s(z)}+B_s(z)||f^{(s) }(z)| \\
&\leq d_2r^{m}(1+o(1))\exp \{ ( 1+\epsilon ) c\delta
(P_s,\theta )r^{n}\} ,
\end{aligned}\label{e4.5}
\end{equation}
where $d_1$, $d_2$ are positive constants. By
\eqref{e4.5},
\begin{equation}
\exp \{ \frac{(1-c)}{2}\delta (P_s,\theta ) r^{n}\}
\leq d_3r^{s},  \label{e4.6}
\end{equation}
where $d_3$  is a positive constant. Hence \eqref{e4.6} is
not possible.

 If $m<s$, by \eqref{e1.6}, \eqref{e4.1} and
\eqref{e4.2}, we obtain for all $z$ with $\arg z=\theta $,
$|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$,
\begin{equation}
\begin{aligned}
&d_{4}r^{s-1}(1-o(1))\exp \{ ( 1-\epsilon ) c_0\delta
(P_s,\theta )r^{n}\} \\
&\leq | A_0(z)e^{P_0(z)}+B_0(z)||f(z)| \\
&\leq \sum_{j=1}^{s-1} |A_j(z)e^{P_j(z)}+B_j(z)|
|f^{(j)}(z)| \\
&\leq d_{5}r^{s-2}(1+o(1))\exp \{ ( 1+\epsilon ) c\delta
(P_s,\theta )r^{n}\},
\end{aligned} \label{e4.7}
\end{equation}
where $d_{4}$, $d_{5}$ are positive constants. By
\eqref{e4.7},
\begin{equation}
\exp \{ \frac{(c_0-c')}{2}\delta ( P_s,\theta
)r^{n}\} \leq \frac{d_{6}}{r}, \label{e4.8}
\end{equation}
where $d_{6}$  is a positive constant. This  contradiction implies that
 if $\max $\{$c_1,\dots c_{s-1}$\}$<c_0$, then every
solution of  \eqref{e1.6} is of infinite order and
satisfies $\sigma _2(f)=n$.


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

Assume $f$ is a transcendental solution of  \eqref{e1.6}.
By Lemma \ref{lem2.1}, there exist a constant $B>0$ and a set
$E_1\subset [ 1,+\infty )$ having finite logarithmic measure such
that for all $z$ satisfying $|z|=r\notin [ 0,1] \cup E_1$, we have
\begin{equation}
\big|\frac{f^{(j)}(z)}{f(z)}\big|\leq Br[ T(2r,f)] ^{k+1}
\quad (j=1,2,\dots,k).
\label{e5.1}
\end{equation}
Set $\beta =\max \{ \sigma (B_j)\; (j=0,\dots ,k-1)\}$.
Suppose that $a_{n,j_1},\dots ,a_{n,j_{m}}$ satisfy
$a_{n,j_{\alpha }}=c_{j_{\alpha }}a_{n,0}$,
$j_{\alpha }\in \{ 1,\dots ,s-1,s+1,\dots k-1\} $,
$\alpha \in \{ 1,\dots ,m\} $, $1\leq m\leq k-2$ and
$\arg a_{n,j}=\theta _s$ for $j\in \{ 1,\dots ,s-1,s+1,\dots ,k-1\}
\setminus \{  j_1,\dots ,j_{m} \} $. Choose a constant
$c$ satisfying $\max \{ c_{j_1},\dots ,c_{j_{m}}\}=c<1$.
We divide the proof into two cases:
 $c<0$ and $0\leq c<1$.

\textbf{Case (a): $c<0$.}
 Since $\theta _0\neq \theta _s$, there is a ray
$\arg z=\theta \in [ 0,2\pi )\setminus H$, where $H=\{ \theta
\in [ 0,2\pi ):\delta (P_0,\theta )=0\text{ or } \delta
(P_s,\theta )=0\} $ such that $\delta ( P_0,\theta )>0$
and $\delta (P_s,\theta )<0$.
 Hence
\begin{gather}
\delta (P_{j_{\alpha }},\theta )=c_{j_{\alpha }}\delta (P_0,\theta
)<0 (\alpha =1,\dots ,m), \label{e5.2}
\\
\delta (P_j,\theta )=|a_{n,j}|\cos (\theta _s+n\theta
)<0,   \label{e5.3}
\end{gather}
where $j\in \{ 1,\dots ,s-1,s+1,\dots ,k-1\} \setminus
\{ j_1,\dots ,j_{m}\} $. By Lemma \ref{lem2.3}, for any given
$\epsilon $ $(0<2\epsilon <\min \{ 1,n-\beta \})$,
there exists a set $E_3\subset [ 1,+\infty )$ having finite
logarithmic measure such that for all $z$ with $\arg z=\theta $,
$|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we
have
\begin{equation}
|A_0(z)e^{P_0(z)}+B_0(z)|\geq (1-o(1))\exp \{ (1-\epsilon
)\delta (P_0,\theta ) r^{n}\} \label{e5.4}
\end{equation}
and
\begin{equation}
\begin{aligned}
|A_j(z)e^{P_j(z)}+B_j(z)|
&\leq \exp \{ (1-\epsilon )\delta (P_j,\theta )r^{n}\}
 +\exp \{ r^{\sigma (B_j)+\frac{\epsilon}{2}}\}   \\
&\leq \exp \{ r^{\sigma (B_j)+\epsilon }\}
 \\
&\leq \exp \{ r^{\beta +\epsilon }\}  ( j=1,\dots ,k-1).
\end{aligned} \label{e5.5}
\end{equation}
We rewrite \eqref{e1.6} as
\begin{equation}
\begin{aligned}
&A_0(z)e^{P_0(z)}+B_0(z)\\
&=\frac{f^{(k)}}{f}+(A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}(z))
\frac{f^{(k-1)}}{f} +\dots \\
&\quad + (A_s(z)e^{P_s(z)}+B_s(z))\frac{f^{(s)}}{f} +\dots
+(A_1(z)e^{P_1(z)}+B_1(z))\frac{ f'}{f}.
\end{aligned} \label{e5.6}
\end{equation}
Hence by \eqref{e5.1} and \eqref{e5.4}-\eqref{e5.6}, we obtain for
all $z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1]\cup E_1\cup
E_3$ and a sufficiently large $r$,
\begin{equation}
\begin{aligned}
&(1-o(1))\exp \{ (1-\epsilon )\delta (P_0,\theta ) r^{n}\} \\
&\leq (1+(k-1)\exp \{r^{\beta +\epsilon }\})Br[ T(2r,f)] ^{k+1} \\
&\leq kBr\exp \{r^{\beta +\epsilon }\}[ T(2r,f)] ^{k+1}.
\end{aligned} \label{e5.7}
\end{equation}
Thus $0<2\epsilon <\min \{ 1,n-\beta \} $ implies
$\sigma (f)=+\infty $ and $\sigma _2(f)\geq n$. By Lemma \ref{lem2.4}, we
have $\sigma _2(f)=n$.



\textbf{Case (b): $0\leq c<1$.}
 Using the same reasoning as above, there
exists a ray $\arg z=\theta \in [ 0,2\pi )\setminus H$, where $H$
is defined as above, such that $\delta (P_0,\theta )>0$, and
$\delta (P_s,\theta )<0$.
Hence
\begin{gather}
\delta (-cP_0,\theta )=-c\delta (P_0,\theta )<0, \delta
((1-c)P_0,\theta )=( 1-c)\delta (P_0,\theta )
>0,  \label{e5.8}
\\
\delta (P_j,\theta )=|a_{n,j}|\cos (\theta _s+n\theta)<0,   \label{e5.9}
\end{gather}
where $j\in \{ 1,\dots ,s-1,s+1,\dots ,k-1\} \setminus
\{ j_1,\dots ,j_{m}\} $,
\begin{gather}
\delta (P_j-cP_0,\theta )<0, j\in \{ 1,\dots ,k-1\}
\setminus \{ j_1,\dots ,j_{m}\}, \label{e5.10}
\\
\delta (P_{j_{\alpha }}-cP_0,\theta )=(c_{j_{\alpha }}-c) \delta
(P_0,\theta )<0 (\alpha =1,\dots ,m). \label{e5.11}
\end{gather}
By Lemma \ref{lem2.3}, for any given $\epsilon $ $(0<2\epsilon <1)$,
there exists a set $E_3\subset [ 1,+\infty )$ having finite
logarithmic measure such that for all $z$ with $\arg z=\theta $,
$|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we
have
\begin{gather}
|A_0(z)e^{(1-c)P_0(z)}|\geq \exp \{ (1-\epsilon
)(1-c)\delta ( P_0,\theta )r^{n}\} , \label{e5.12}
\\
|e^{-cP_0(z)}|\leq \exp \{ -(1-\epsilon )c\delta
(P_0,\theta )r^{n}\} <M, \label{e5.13}
\\
|B_j(z)e^{-cP_0(z)}|\leq \exp \{ -(1-\epsilon )c\delta
(P_0,\theta )r^{n}\} <M , \label{e5.14}
\\
|A_j(z)e^{P_j(z)-cP_0(z)}|\leq \exp \{ (1-\epsilon
)\delta ( P_j-cP_0,\theta ) r^{n}\} <M ,
\label{e5.15}
\end{gather}
where $j=1,\dots ,k-1$, and $M$ is a positive constant.
We can rewrite \eqref{e1.6}  as
\begin{equation}
\begin{aligned}
A_0(z)e^{(1-c)P_0(z)}
&=-B_0(z)e^{-cP_0( z) }+e^{-cP_0(z)}\frac{f^{(k)}}{f}\\
&\quad + (A_{k-1}(z)e^{P_{k-1}(z)-cP_0(z)
}+B_{k-1}(z)e^{-cP_0(z)})\frac{f^{(k-1)}}{ f}+\dots \\
&\quad + (A_s(z)e^{P_s(z)-cP_0(z)}+B_s(z)e^{-cP_0(z) })
\frac{f^{(s)}}{f}+\dots \\
&\quad + (A_1(z)e^{P_1(z)-cP_0(z)}+B_1(z)e^{-cP_0(z) })
\frac{f'}{f}.
\end{aligned} \label{e5.16}
\end{equation}
By \eqref{e5.1}, \eqref{e5.12}-\eqref{e5.16}, for all $z$ with
$| z| =r\notin [ 0,1] \cup E_1\cup E_3$ and a sufficiently large
$r$, we have
\begin{equation}
\exp \{ (1-\epsilon )(1-c)\delta ( P_0,\theta )
r^{n}\} \leq M'r[ T(2r,f)] ^{k+1}, \label{e5.17}
\end{equation}
where $M'$  is a positive constant. Thus $0<2\epsilon <1$ and
\eqref{e5.17} implie $\sigma ( f) =+\infty $ and
$\sigma _2(f)\geq n$. By Lemma \ref{lem2.4}, we have $\sigma _2(f)=n$.
\smallskip

Now we prove that equation \eqref{e1.6} cannot have a nonzero
polynomial solution. Let $f(z)$ be a nonzero polynomial solution
of  \eqref{e1.6} with $\deg f(z)=q$. Suppose first that
$\max \{ c_{j_1},\dots ,c_{j_{m}}\} =c<0$. Using the
same reasoning as above, there is a ray $\arg z=\theta \in [
0,2\pi )\setminus H$, where $H$ is defined as above, such that
$\delta (P_0,\theta )>0$, and $\delta (P_s,\theta )<0$. By
Lemma \ref{lem2.3}, for any given $\epsilon $
$( 0<2\epsilon <\min \{1,n-\beta \} )$, there exists a set
$E_3\subset [1,+\infty )$ having finite logarithmic measure such
that for all $z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1 ] \cup E_3$
and a sufficiently large $r$, we have \eqref{e5.4} and \eqref{e5.5}.

 By \eqref{e1.6}, \eqref{e5.4} and \eqref{e5.5}, for all
$z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1 ] \cup E_3$ and a
sufficiently large $r$, we have
\begin{equation}
\begin{aligned}
\gamma _1r^{q}(1-o(1))\exp \{ ( 1-\epsilon ) \delta
(P_0,\theta )r^{n}\}
&\leq | A_0(z)e^{P_0(z)}+B_0(z)||f( z) | \\
&\leq k\gamma _2r^{q-1}\exp \{r^{\beta +\epsilon }\}.
\end{aligned}\label{e5.18}
\end{equation}
where $\gamma _1$  and $\gamma _2$  are positive constants.
From \eqref{e5.18},
\begin{equation}
\exp \{ (1-\epsilon )\delta (P_0,\theta ) r^{n}\}
\leq \frac{\gamma _3}{r},  \label{e5.19}
\end{equation}
where $\gamma _3$  is a positive constant. This is a
contradiction. Suppose now that $0\leq c<1$. Using the same
reasoning as above, there is a ray $\arg z=\theta \in [ 0,2\pi
)\setminus H$, where $H$ is defined as above, such that $\delta
(P_0,\theta )>0$, and $\delta (P_s,\theta )<0$.
By Lemma \ref{lem2.3}, for any $\epsilon $ $(0<2\epsilon <1)$, there exists
a set $E_3\subset [ 1,+\infty )$ having finite logarithmic measure
such that for all $z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1 ]
\cup E_3$ and a sufficiently large $r$, we have
\eqref{e5.12}-\eqref{e5.15}.

 By \eqref{e1.6}, \eqref{e5.12}-\eqref{e5.15}, for all
$z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1] \cup E_3$ and a
sufficiently large $r$, we have
\begin{equation}
\begin{aligned}
&\gamma _{4}r^{q}\exp \{ (1-\epsilon )( 1-c)\delta
(P_0,\theta )r^{n}\} \\
&\leq | A_0(z)e^{(1-c)P_0(z)}||f( z) |\\
&\leq |B_0(z)e^{-cP_0(z)}||f(z)|+|e^{-cP_0( z)}| |f^{(k)}(z)|\\
&\quad + |A_{k-1}(z)e^{P_{k-1}(z)-cP_0(z) }+B_{k-1}(z)
e^{-cP_0(z)}||f^{(k-1)}(z) |\\
&\quad +\dots +|A_1(z)e^{P_1(z)-cP_0(z) }+B_1(z) e^{-cP_0(z)}||f'(z) |\\
&\leq \gamma _{5}r^{q},
\end{aligned} \label{e5.20}
\end{equation}
where $\gamma _{4}$  and $\gamma _{5}$  are positive constants.
 From \eqref{e5.20}, we obtain
for $|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$,
\begin{equation}
\exp \{ (1-\epsilon )(1-c)\delta ( P_0,\theta )
r^{n}\} \leq \frac{\gamma _{5}}{\gamma _{4}}\,. \label{e5.21}
\end{equation}
This is a contradiction; hence  \eqref{e1.6} cannot
have a nonzero polynomial solution.


If $\arg a_{n,j}=\theta _s$ $(j=1,\dots ,s-1,s+1,\dots ,k-1)$,
then $\arg a_{n,j}\neq \arg a_{n,0}$ $(j=1,\dots ,k-1)$ and by
Theorem \ref{thmC}, it follows that every solution $f$ $(\not\equiv 0)$ of
 \eqref{e1.6} is of infinite order and satisfies $\sigma_2(f) =n$.



\subsection*{Proof of Theorem \ref{thm1.4}}

Assume $f$ is a transcendental solution of  \eqref{e1.6}.
By Lemma \ref{lem2.1}, there exist a constant $B>0$ and a set
$E_1\subset [ 1,+\infty )$ having finite logarithmic measure such
that for all $z$ satisfying $|z|=r\notin [ 0,1]\cup E_1$, we have
\begin{gather}
|\frac{f^{(j)}(z)}{f^{(d)}(z)}|\leq Br [ T(2r,f)]
^{j-d+1}\quad (j=d+1,\dots,k)  \label{e6.1}
\\
|\frac{f^{(j)}(z)}{f(z)}|\leq Br[ T(2r,f)] ^{j+1}\quad
(j=1,2,\dots,d-1).
\label{e6.2}
\end{gather}
By Lemma \ref{lem2.2}, there exists a set $E_2\subset [ 1,+\infty )$
having finite logarithmic measure such that for all $z$
satisfying $|z|=r\notin [ 0,1] \cup E_2$ and $|f(z)| =M(r,f)$,
we have
\begin{equation}
|\frac{f(z)}{f^{(d)}(z)} |\leq 2r^{d}. \label{e6.3}
\end{equation}
Set $\beta =\max \{ \sigma (B_j)\; (j=0,\dots ,k-1)\}$.
 Suppose that $a_{n,j_1},\dots ,a_{n,j_{m}}$ satisfy
$a_{n,j_{\alpha }}=c_{j_{\alpha }}a_{n,d}$, $j_{\alpha }\in
\{ 0,\dots ,k-1\} \setminus \{ d,s\} $,
$\alpha \in \{ 1,\dots ,m\} $, $1\leq m\leq k-2$ and
$\arg a_{n,j}=\theta _s$ for $j\in \{ 0,\dots ,k-1\}
\setminus \{ d,s,j_1,\dots ,j_{m}\} $. Choose a
constant $c$ satisfying $\max \{ c_{j_1},\dots
,c_{j_{m}}\} =c<1$. We divide the proof into two cases:
 $c<0$ and $0\leq c<1$.

\textbf{Case (a): $c<0$.}
 Since $\theta _d\neq \theta _s$, there is a
ray $\arg z=\theta \in [ 0,2\pi )\setminus H$, where
$H=\{ \theta \in [ 0,2\pi ):\delta (P_d,\theta )=0\text{ or } \delta
(P_s,\theta )=0\} $ such that $\delta ( P_d,\theta )>0$
and $\delta (P_s,\theta )<0$.
Hence
\begin{gather}
\delta (P_{j_{\alpha }},\theta )=c_{j_{\alpha }}\delta
(P_d,\theta )<0 \quad (\alpha =1,\dots ,m), \label{e6.4}
\\
\delta (P_j,\theta )=|a_{n,j}|\cos (\theta _s+n\theta
)<0, \quad j\in \{ 0,\dots ,k-1\} \setminus \{
d,s,j_1,\dots ,j_{m}\} . \label{e6.5}
\end{gather}
By Lemma \ref{lem2.3}, for any  $\epsilon $ $(0<2\epsilon <\min
\{ 1,n-\beta \} )$, there exists a set $E_3\subset [
1,+\infty )$ having finite logarithmic measure such that for all
$z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1] \cup E_3$ and a
sufficiently large $r$, we have
\begin{equation}
|A_d(z)e^{P_d(z)}+B_d(z)|\geq (1-o(1))\exp \{
(1-\epsilon )\delta (P_d,\theta ) r^{n}\} \label{e6.6}
\end{equation}
and
\begin{equation}
\begin{aligned}
|A_j(z)e^{P_j(z)}+B_j(z)|
&\leq \exp \{ (1-\epsilon
)\delta (P_j,\theta )r^{n}\} +\exp \{ r^{\sigma
(B_j)+\frac{\epsilon }{2}}\}   \\
&\leq \exp \{ r^{\sigma (B_j)+\epsilon }\}
 \\
&\leq \exp \{ r^{\beta +\epsilon }\}  (j\neq d).
\end{aligned}\label{e6.7}
\end{equation}
By \eqref{e1.6}, we have
\begin{equation}
\begin{aligned}
&A_d(z)e^{P_d(z)}+B_d(z)\\
&=\frac{f^{(k)}}{f^{( d)
}}+\big(A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}(z)\big)\frac{
f^{(k-1)}}{f^{(d)}}+\dots \\
&\quad + \big(A_{d+1}(z)e^{P_{d+1}(z)}+B_{d+1}(z)\big)
 \frac{ f^{(d+1)}}{f^{(d)}}\\
&\quad +\big(A_{d-1}(z)e^{P_{d-1}(z)}
+B_{d-1}(z)\big)\frac{f^{(d-1)}}{f}\frac{f}{f^{(d)}}
 +\dots \\
&\quad +(A_1(z)e^{P_1(z)}+B_1(z))\frac{
f'}{f}\frac{f}{f^{(d)}}+( A_0(z)e^{P_0(z)}+B_0(z))
\frac{f}{f^{(d)}}\, .
\end{aligned} \label{e6.8}
\end{equation}
Hence by \eqref{e6.1}-\eqref{e6.3} and \eqref{e6.6}-\eqref{e6.8}, we
get for all $z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1]\cup
E_1\cup E_2\cup E_3$, $|f(z)|=M(r,f)$ and a sufficiently large
$r$,
\begin{equation}
(1-o(1))\exp \{ (1-\epsilon )\delta (P_d,\theta )r^{n}\}
\leq M_1r^{d+1}\exp \{r^{\beta +\epsilon }\}[ T(2r,f)] ^{k+1}, \label{e6.9}
\end{equation}
where $M_1$  is a positive constant. Thus $0<2\epsilon <\min
\{ 1,n-\beta \} $ implies $\sigma (f)=+\infty $ and
$\sigma _2(f)\geq n$. By Lemma \ref{lem2.4}, we have $\sigma _2(f) =n$.


\textbf{Case (b): $0\leq c<1$.}
 Using the same reasoning as above, there
exists a ray $\arg z=\theta \in [ 0,2\pi )\setminus H$, where $H$
is defined as above, such that $\delta (P_d,\theta )>0$, and
$\delta (P_s,\theta )<0$.
Hence
\begin{gather}
\delta (-cP_d,\theta )=-c\delta (P_d,\theta )<0, \delta
((1-c)P_d,\theta )=( 1-c)\delta (P_d,\theta )
>0,  \label{e6.10}
\\
\delta (P_j,\theta )=|a_{n,j}|\cos (\theta _s+n\theta
)<0,\quad j\in \{ 0,\dots ,k-1\} \setminus \{
d,s,j_1,\dots ,j_{m}\} , \label{e6.11}
\\
\delta (P_j-cP_d,\theta )<0 \quad j\in \{ 0,\dots ,k-1\}
\setminus \{ d,j_1,\dots ,j_{m}\}, \label{e6.12}
\\
\delta (P_{j_{\alpha }}-cP_d,\theta )=(c_{j_{\alpha }}-c) \delta
(P_d,\theta )<0 \quad (\alpha =1,\dots ,m). \label{e6.13}
\end{gather}
By Lemma \ref{lem2.3}, for any given $\epsilon $ $(0<2\epsilon <1)$,
there exists a set $E_3\subset [ 1,+\infty )$ having finite
logarithmic measure such that for all $z$ with $\arg z=\theta $,
$|z|=r\notin [ 0,1] \cup E_3$ and a sufficiently large $r$, we
have
\begin{gather}
|A_d(z)e^{(1-c)P_d(z)}|\geq \exp \{ (1-\epsilon
)(1-c)\delta ( P_d,\theta )r^{n}\} , \label{e6.14}
\\
|e^{-cP_d(z)}|\leq \exp \{ -(1-\epsilon )c\delta
(P_d,\theta )r^{n}\} <M_2, \label{e6.15}
\\
|B_j(z)e^{-cP_d(z)}|\leq \exp \{ -(1-\epsilon )c\delta
(P_d,\theta )r^{n}\} <M_2 \quad (j=0,\dots ,k-1).
\label{e6.16}
\\
|A_j(z)e^{P_j(z)-cP_d(z)}|\leq \exp \{ (1-\epsilon
)\delta ( P_j-cP_d,\theta ) r^{n}\} <M_2 \quad (j\neq d),
\label{e6.17}
\end{gather}
where $M_2$  is a positive constant. We can rewrite \eqref{e1.6}
as
\begin{equation}
\begin{aligned}
&A_d(z)e^{(1-c)P_d(z)}\\
&=-B_d(z) e^{-cP_d(z)}+e^{-cP_d(z)}\frac{f^{(k)}}{ f^{(d)}}\\
&\quad + (A_{k-1}(z)e^{P_{k-1}(z)-cP_d(z)
}+B_{k-1}(z)e^{-cP_d(z)})\frac{f^{(k-1)}}{ f^{(d)}}+\dots \\
&\quad + (A_{d+1}(z)e^{P_{d+1}(z)-cP_d(z)}+B_{d+1}(z) e^{-cP_d(z)
})\frac{f^{(d+1)}}{f^{(d)}}\\
&\quad + (A_{d-1}(z)e^{P_{d-1}(z)-cP_d(z)}+B_{d-1}(
z)e^{-cP_d(z)})\frac{f^{(d-1)}}{f}\frac{f}{ f^{(d)}} \\
&\quad +\dots +(A_1(z)e^{P_1(z)-cP_d(z)}+B_1(z)
e^{-cP_d(z)})\frac{f'}{f}\frac{f}{ f^{(d)}}\\
&\quad + (A_0(z)e^{P_0(z)-cP_d(z)}+B_0(z)e^{-cP_d(z) })
\frac{f}{f^{(d)}}.
\end{aligned} \label{e6.18}
\end{equation}
By \eqref{e6.1}-\eqref{e6.3} and \eqref{e6.14}-\eqref{e6.18}, for all
$z$ with $\arg z=\theta $, $|z|=r\notin [ 0,1] \cup E_1\cup
E_2\cup E_3$, $|f(z)|=M(r,f)$ and a sufficiently large $r$, we
have
\begin{equation}
\exp \{ (1-\epsilon )(1-c)\delta ( P_d,\theta )
r^{n}\} \leq M_3r^{d+1}[ T(2r,f)] ^{k+1}, \label{e6.19}
\end{equation}
where $M_3$ is a positive constant. Thus $0<2\epsilon <1$
implies $\sigma (f)=+\infty $ and $\sigma _2( f)\geq n$. By
Lemma \ref{lem2.4}, we have $\sigma _2( f)=n$.
\smallskip

If $\arg a_{n,j}=\theta _s$ $(j\neq d,s)$, then
$\arg a_{n,j}\neq \arg a_{n,d}$ $(j\neq d)$ and by Theorem \ref{thm1.1}, it
follows that every transcendental solution $f$ of equation
\eqref{e1.6} is of infinite order and satisfies $\sigma_2(f)=n$.

\begin{thebibliography}{00}

\bibitem{b1} B. Belaidi;
\emph{Some precise estimates
of the hyper order of solutions of some complex linear
differential equations}, JIPAM. J. Inequal. Pure Appl. Math. 8
(2007), no. 4, Article 107, 14 pp.

\bibitem{c1} Z. X. Chen;
\emph{On the hyper-order of solutions of some second order
linear differential equations},
Acta. Math. Sinica Engl. Ser., 18 (1) (2002), 79-88.

\bibitem{c2} Z. X. Chen;
\emph{The growth of solutions of $f''+e^{-z}f'+Q(z) f=0$,
where the order $(Q)=1$}, Sci. China
Ser. A, 45 (2002), 290-300.

\bibitem{c3} Z.-X. Chen and K. H. Shon;
\emph{On the growth of solutions of a class of higer order
linear differential equations},
Acta. Mathematica Scientia, 24 B (1) (2004), 52-60.

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

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

\bibitem{h1} W. K. Hayman;
\emph{Meromophic functions}, Clarendon Prss, Oxford, 1964.

\bibitem{k1} K. H. Kwon;
\emph{Nonexistence of finite order solutions of certain second
linear differential equations},
Kodai. Math. J., 19 (1996), 378-387.

\bibitem{m1} A. I. Markushevich;
\emph{Theory of functions of a complex variable}
(Translated by R. A. Sibnerman), Vol 2, Prentice-Hall,
Englewood Cliffs, New Jersy, 1965.

\bibitem{t1} J. Tu and C. F. Yi;
\emph{On the growth of solutions of a class of higher order
linear differential equations with coefficients having the same order},
J. Math. Appl. 340 (2008), no. 1 487-497.

\bibitem{y1} H. X. Yi and C. C. yang;
\emph{The uniqueness theory of meromorphic functions}, Science Press,
Beijing, 1995.

\end{thebibliography}

\end{document}