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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 157, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/157\hfil
Entire coefficients having the same order and type]
{Linear differential equations with entire
coefficients having the same order and type}

\author[N. Berrighi, S. Hamouda \hfil EJDE-2011/157\hfilneg]
{Nacera Berrighi, Saada Hamouda}  % in alphabetical order

\address{Nacera Berrighi \newline
Laboratory of Pure and Applied Mathematics\\
University of Mostaganem, B.P. 227 Mostaganem, Algeria}
\email{n\_berrighi@yahoo.fr}

\address{Saada Hamouda \newline
Laboratory of Pure and Applied Mathematics\\
University of Mostaganem, B.P. 227 Mostaganem, Algeria}
\email{hamouda\_saada@yahoo.fr}

\thanks{Submitted October 20, 2011. Published November 21, 2011.}
\subjclass[2000]{34M10, 30D35}
\keywords{Linear differential equations;
 growth of solutions; hyper-order}

\begin{abstract}
 In this article, we study the growth of solutions to the
 differential equation
 \begin{align*}
 &f^{k}+(A_{k-1}(z)e^{P_{k-1}(z)}e^{\lambda z^m}
 +B_{k-1}(z))f^{k-1}+\dots \\
 &+(A_0(z)e^{P_0(z)}  e^{\lambda z^m}+B_0(z))f=0,
 \end{align*}
 where $\lambda \in \mathbb{C}^{\ast}$, $m\geq 2$ is an
 integer and $\max_{j=0,\dots ,k-1}\{ \deg P_j(z)\} <m$,
 $A_j,B_j$ $(j=0,\dots ,k-1)$ are entire functions of orders
 less than $m$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction and statement of results}

Throughout this paper, we assume that the reader is familiar with the
fundamental results and the standard notation of the Nevanlinna value
distribution theory (see \cite{haym}). In addition, we
use the notation $\sigma _2(f)$ to denote the hyper-order
of nonconstant entire function $f$; that is,
\[
\sigma _2(f)=\limsup_{r\to +\infty } \frac{\log \log T(r,f)}{\log r}
=\limsup_{r\to +\infty} \frac{\log \log \log M(r,f)}{\log r},
\]
where $T(r,f)$ is the Nevanlinna characteristic
 of $f$ and $M(r,f)=\max_{|z|=r}|f(z)|$ (see \cite{Yi-Yang}).

We define the linear measure of a set $E\subset [0,2\pi )$ by
$m(E)=\int_0^{+\infty }\chi _{E}(t)dt$
and the logarithmic measure of a set $F\subset [1,+\infty )$ by
$lm(F)=\int_1^{+\infty }\frac{\chi _{F}(
t)}{t}dt$, where $\chi _{H}(t)$ is the characteristic
function of a set $H$.

Several authors have studied the particular differential equations
\begin{equation}
f''+e^{-z}f'+Q(z)f=0,  \label{eq0}
\end{equation}
(see \cite{ozaw,frei,gund2,lang}).
Gundersen \cite{gund2} proved that if
$\deg Q(z)\neq 1$, then every nonconstant solution of
\eqref{eq0} is of infinite order. Chen  considered the case
$Q(z)=h(z)e^{bz}$, where $h(z)$ is
nonzero polynomial and $b\neq -1$, see \cite{chen1};
more precisely, he proved that every nontrivial solution
$f$ of \eqref{eq0} satisfies $\sigma _2(f)=1$.
The same paper contains a discussion about more general
equations of the type
\begin{equation}
f''+A_1(z)e^{az}f'+A_0(
z)e^{bz}f=0,  \label{eq1}
\end{equation}
where $A_1\not\equiv 0,A_0\not\equiv 0$, are entire
functions of order less than $1$, and $a,b$ are complex constants.
He proved that if $ab\neq 0$
and $\arg a\neq \arg b$ or $a=cb$ $(0<c<1)$, then every
solution $f(z)\not\equiv 0$ of \eqref{eq1} is of
infinite order. He also proved the following result.

\begin{theorem}[\cite{chen1}] \label{thmm1}
Let $A_j(z)$ $(\not\equiv 0)$, $
D_j(z)$ $(j=0,1)$ be entire functions with $
\sigma (A_j)<1$, $\sigma (B_j)<1$ $(j=0,1)$, $a,b$ be 
complex constants such that $ab\neq 0$ and $\arg
a\neq \arg b$ or $a=cb$ $(0<c<1)$. Then every solution
$f(z)\not\equiv 0$ of
\[
f''+(A_1(z)e^{az}+D_1)f'+(A_0(z)e^{bz}+D_0)f=0,
\]
is of infinite order.
\end{theorem}

In another paper, Chen and Shon \cite{chen2}  Proved the following result.

\begin{theorem} \label{thmm2}
 Let $P(z)=\sum_{i=0}^{n}a_iz^i$ and
$Q(z)=\sum_{i=0}^{n}b_iz^i$ be nonconstant
polynomials where $a_i,b_i$ $(i=0,1,\dots ,n)$ are complex
numbers, $a_{n}\neq 0$, $b_{n}\neq 0$. Let $A_1(z)\not\equiv
0 $ and $A_0(z)\not\equiv 0$ be entire functions. Suppose
that either (i) or (ii) below, holds:
\begin{itemize}
\item[(i)] $\arg a_{n}\neq \arg b_{n}$ or $a_{n}=cb_{n}$
$(0<c<1)$ $\sigma (A_j)<n$ $(j=0,1)$
\item[(ii)] $a_{n}=cb_{n}$ $(c>1)$ and $\deg (P-cQ)=m\geq
1$, $\sigma (A_j)<m$ $(j=0,1)$.
\end{itemize}
Then every solution $f(z)\not\equiv 0$ of the differential
equation
\[
f''+A_1(z)e^{P(z)}f'+A_0(z)e^{Q(z)}f=0,
\]
is of infinite order with $\sigma _2(f)=n$.
\end{theorem}

Hamouda and Belaidi \cite{ham-bel} investigated the linear
differential equation
\[
w^{(n)}+e^{az^m}w'+Q(z)w=0
\]
and some related extensions.

In this paper, we investigate the  differential equation
\begin{equation}  \label{eq2}
\begin{split}
&f^{k}+(A_{k-1}(z)e^{P_{k-1}(z)}e^{\lambda
z^m}+B_{k-1}(z))f^{k-1}+\dots\\
&+(A_0(z)e^{P_0(z)}e^{\lambda z^m}+B_0(z))f=0,
\end{split}
\end{equation}
where $\lambda \in \mathbb{C}^{\ast}$, $m\geq 2$ is an integer and
$\max_{j=0,\dots ,k-1}\{ \deg P_j(z)\} <m$. We
obtain the following results.

\begin{theorem}\label{thm1} Let $\lambda \in \mathbb{C}^{\ast}$,
$m\geq 2$ is an integer and $P_0(z),\dots ,P_{k-1}(z)$
be non constant polynomials such that
$\max_{j=0,\dots ,k-1}\{ \deg P_j(z)\}<m$;
$A_j(z)$ $(\not\equiv 0)$,
$B_j(z)$ $(j=0,\dots ,k-1)$ be entire functions
such that $\sigma (A_j)< \deg P_j(z)$, $\sigma (B_j)<m$ $(j=0,\dots ,k-1)$.
Suppose that there exist $\theta _1<\theta_2$ such that
$\delta (\lambda z^m,\theta )>0$,
$\delta (P_0,\theta )>0$ and $\delta (P_j,\theta )<0$
$(j=1,\dots ,k-1)$ for all
$\theta \in (\theta _1,\theta_2)$. Then every non trivial
solution $f$ of \eqref{eq2} is of
infinite order with $n\leq \sigma _2(f)\leq m$,
where $n=\deg P_0$.
\end{theorem}

\begin{corollary}\label{cor1}
Let $P_j(z)=\sum_{i=0}^{n}a_{i,j}z^i$
$(j=0,\dots ,k-1)$ be non constant polynomials where
$a_{i,j}$ are complex numbers such that $a_{n,j}\neq 0$
$(j=0,\dots ,k-1)$, $\arg a_{n,j}=\arg a_{n,1}$
$(j=2,\dots ,k-1)$ and $\arg a_{n,1}\neq \arg a_{n,0}$;
$A_j(z)$ $(\not\equiv 0)$, $B_j(z)$, $(j=0,\dots ,k-1)$
be entire functions such that $\sigma (A_j)<n$,
$\sigma (B_j)<m$ $(j=0,\dots ,k-1)$.
Then every non trivial solution $f$ of \eqref{eq2} is of
infinite order with $n\leq \sigma _2(f)\leq m$.
\end{corollary}

 Now we give examples for Theorem \ref{thm1} for  cases other than
Corollary \ref{cor1}.

\begin{example} \label{examp1} \rm
From Theorem \ref{thm1}, every non trivial solution $f$ of the
differential equation
\begin{align*}
&f'''+(A_2(z) e^{z^{3}}e^{z^{4}}+B_2(z))f''+(
A_1(z)e^{z^{2}}e^{z^{4}}+B_1(z))f'\\
&+ (A_0(z)e^{z}e^{z^{4}}+B_0(z))f=0,
\end{align*}
is of infinite order with $1\leq \sigma _2(f)\leq 4$.
We cane take $(\theta _1,\theta _2)\subset
(\pi/3,\pi /2)\cup (3\pi/2,5\pi/3)$
\end{example}

\begin{example} \label{examp2} \rm
Every non trivial solution $f$ of the differential equation
\begin{align*}
&f'''+(A_2(z) e^{z}e^{z^{3}}+B_2(z))f''+(
A_1(z)e^{(i+1)z^{2}}e^{z^{3}}+B_1(z))f'\\
&+(A_0(z)e^{z^{2}}e^{z^{3}}+B_0(z))f=0,
\end{align*}
is of infinite order with $2\leq \sigma _2(f)\leq 3$.
Here we can take $(\theta _1,\theta _2)\subset
(3\pi/4,5\pi/6)$.
\end{example}

\begin{theorem}\label{thm2}
Let $P_j(z)=\sum_{i=0}^{n}a_{i,j}z^i$
$(j=0,\dots ,k-1)$ be non constant polynomials where $a_{i,j}$ are
complex numbers such that $a_{n,0}\neq 0$ and
$a_{n,j}=c_ja_{n,0}$ $(0<c_j<1)$ $(j=1,\dots ,k-1)$;
$A_j(z)$ $(\not\equiv 0)$, $B_j(z)$,
$(j=0,\dots ,k-1)$ be entire functions such that
$\sigma (A_j)<n$, $\sigma (B_j)<m$ $(j=0,\dots ,k-1)$.
Then every non trivial solution $f$ of \eqref{eq2} is of infinite
order with $n\leq \sigma _2(f)\leq m$.
\end{theorem}

By combining Corollary \ref{cor1} and Theorem \ref{thm2}, we can
obtain the following corollary.

\begin{corollary} \label{cor2}
Let $P_j(z)=\sum_{i=0}^{n}a_{i,j}z^i$
$(j=0,\dots ,k-1)$ be non constant polynomials where $a_{i,j}$ are
complex numbers such that $a_{n,0}\neq 0$. Suppose that there
exists $s\in \{1,\dots ,k-1\}$ such that
$\arg a_{n,s}\neq \arg a_{n,0}$ and
for all $j\neq 0,s, a_{n,j}$ satisfies either
$a_{n,j}=c_ja_{n,0}$ $(0<c_j<1)$ or $\arg a_{n,j}=\arg a_{n,s}$.
Then every non trivial solution $f$ of \eqref{eq2} is of infinite
order with $n\leq \sigma_2(f)\leq m$.
\end{corollary}

Now we investigate cases when $a_{n,j}$ have distinct arguments or
$a_{n,j}=c_ja_{n,0}$ $(c_j>1)$ and obtain the following
results.

\begin{theorem}\label{thm3}
Let $P_j(z)=\sum_{i=0}^{n}a_{i,j}z^i$
$(j=0,\dots ,k-1)$ be non constant polynomials where $a_{i,j}$ are
complex numbers such that $a_{n,0}\neq 0$. Suppose that there
exists $s\in \{ 1,\dots ,k-1\} $ such that
\begin{equation}
\arg (a_{n,j}-a_{n,s})=\varphi \neq \arg (
a_{n,0}-a_{n,s})\quad \text{\ for all }j\neq 0,s.  \label{eg1}
\end{equation}
Then every non trivial solution $f$ of \eqref{eq2} is of infinite
order with $n\leq \sigma _2(f)\leq m$.
\end{theorem}

\begin{theorem}\label{thm4}
Let $P_j(z)=\sum_{i=0}^{n}a_{i,j}z^i$ $(j=0,\dots ,k-1)$
be non constant polynomials where $a_{i,j}$ are
complex numbers such that $a_{n,0}\neq 0$. Suppose that there
exists $s\in \{ 1,\dots ,k-1\} $ such that
\begin{equation}
a_{n,j}-a_{n,s}=c_j(a_{n,0}-a_{n,s})\ (0<c_j<1)
\label{eg2}
\end{equation}
for all $j\neq 0,s$. Then every non trivial solution $f$
of \eqref{eq2} is of infinite order with $n\leq \sigma _2(f)\leq m$.
\end{theorem}

By combining Theorem \ref{thm3} and Theorem \ref{thm4},
we obtain the following corollary.

\begin{corollary}\label{cor3}
Let $P_j(z)=\sum_{i=0}^{n}a_{i,j}z^i$ $(j=0,\dots ,k-1)$ be
non constant polynomials where $a_{i,j}$ are
complex numbers such that $a_{n,0}\neq 0$. Suppose that there exists
$s\in\{ 1,\dots ,k-1\} $ such that $\arg (a_{n,j}-a_{n,s})
\neq \arg (a_{n,0}-a_{n,s})$ and for $j\neq 0,s$ we have either
\eqref{eg1} or \eqref{eg2}. Then every non trivial solution $f$
of \eqref{eq2} is of infinite order with $n\leq \sigma _2(f)\leq m$.
\end{corollary}

\begin{theorem} \label{thm5}
If $P_j(z)$ and $A_j(z)$ $(j=0,\dots ,k-1)$ satisfy the conditions 
of one of our previous theorems or corollaries, 
then every non trivial solution $f$ of
\begin{equation}
f^{k}+A_{n-1}(z)e^{P_{k-1}(z)}e^{\lambda
z^m}f^{k-1}+\dots +A_0(z)e^{P_0(z)}e^{\lambda
z^m}f=0,  \label{eq3}
\end{equation}
is of infinite order with $\sigma _2(f)=n\ $or $\sigma
_2(f)=m$.
\end{theorem}

\section{Preliminary lemmas}

We need the following lemmas for our proofs.

\begin{lemma}[\cite{gund1}] \label{lem1}
Let $f(z)$ be a transcendental meromorphic function and
$\alpha >1$. There exist a set $E\subset [0,2\pi )$ that has
linear measure zero and a constant $M>0$ that depends
only on $\alpha $ such that  for any
$\theta \in [0,2\pi )\backslash E $ there exists a constant
$R_0=R_0(\theta )>1$ such that for all $z$  satisfying
$\arg z=\theta $ and $|z|=r>R_0$, we have
\[
|\frac{f^{(k)}(z)}{f(z)}
|\leq M\big[T(\alpha r,f)\frac{(\log
^{\alpha }r)}{r}\log T(\alpha r,f)\big] ^{k},\quad
k\in \mathbb{N}.
\]
\end{lemma}

\begin{lemma}[\cite{gund1}]\label{lem1b}
Let $f(z)$ be a transcendental meromorphic function of finite order
$\sigma $, and let $\varepsilon >0$ be a given constant.
Then there exists a set $E\subset [0,2\pi )$ of linear measure
zero such that for all $z=re^{i\theta }$ with $|z|$ sufficiently
large and $\theta \in [0,2\pi )\backslash E$, and for all $k,j$,
$0\leq j\leq k$, we have
\[
|\frac{f^{(k)}(z)}{f^{(j)}(z)}|\leq |z|^{(k-j)
(\sigma -1+\varepsilon )}.
\]
\end{lemma}

Using the Wiman-Valiron theory, we can easily prove the following
lemma (see \cite{chen1}).

\begin{lemma}\label{lem2}
Let $A,B$ be entire functions of finite order. If $f$ is a
solution of the differential equation
\[
f^{(k)}+A_{k-1}f^{(k-1)}+\dots +A_1f'+A_0f=0,
\]
then $\sigma _2(f)\leq \max_{j=0,\dots ,k-1}\{ \sigma(Aj)\} $.
\end{lemma}

\begin{lemma}[\cite{chen1}]\label{lem3}
Let $P(z)=a_{n}z^{n}+\dots $,
$(a_{n}=\alpha +i\beta \neq 0)$ be a
polynomial with degree $n\geq 1$ and
$A(z)$ $(\not\equiv 0)$ be entire function with
$\sigma (A)<n$. Set $f(z)=A(z)e^{P(z)}$,
$z=re^{i\theta }$, $\delta (P,\theta )=\alpha \cos n\theta
-\beta \sin n\theta$.  Then for any given
$\varepsilon >0$,  there exists a set
$H\subset [0,2\pi)$ that has linear measure zero, such that for any
$\theta \in [0,2\pi )\backslash H$,  where
$H=\{ \theta \in [0,2\pi ):\delta (P,\theta )=0\} $
 is a finite set, there is $R>0$ such that for $|z|=r>R$,
we have
\begin{itemize}
\item[(i)] if $\delta (P,\theta )>0$,  then
\[
\exp \{ (1-\varepsilon )\delta (P,\theta )
r^{n}\} \leq |f(z)|\leq \exp \{
(1+\varepsilon )\delta (P,\theta )r^{n}\} ,
\]
\item[(ii)]  if $\delta (P,\theta )<0$,  then
\[
\exp \{ (1+\varepsilon )\delta (P,\theta )
r^{n}\} \leq |f(z)|\leq \exp \{
(1-\varepsilon )\delta (P,\theta )r^{n}\} .
\]
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{chen2}]\label{lem5}
Let $f(z)$ be a entire function with $\sigma (f)=+\infty $ and
$\sigma _2(f)=\alpha <+\infty $, let a set $E_2\subset [1,+\infty )$
has finite logarithmic measure. Then there exists a sequence
$\{z_{p}=r_{p}e^{i\theta _{p}}\} $ such that $f(z_{p})
=M(r_{p},f)$, $\theta _{p}\in [0,2\pi )$,
$\lim_{p\to \infty }\theta _{p}=\theta _0\in [0,2\pi )$,
$r_{p}\notin E_2$, and for any given $\epsilon >0$, as
$r_{p}\to \infty $, we have
\[
\exp \{ r_{p}^{\alpha -\epsilon }\} \leq \nu (r_{p})
\leq \exp \{ r_{p}^{\alpha +\epsilon }\} ,
\]
where $\nu (r)$ is the central index of $f$.
\end{lemma}

\section{Proofs of theorems}

\begin{proof}[Proof of theorem \protect\ref{thm1}]
From \eqref{eq2}, we obtain
\begin{equation} \label{f1}
\begin{split}
&|A_0(z)e^{P_0(z)}+B_0(z)e^{-\lambda z^m}|\\
&\leq |e^{-\lambda z^m}||\frac{f^{(k)}}{f}|+
\sum_{j=1}^{k-1}|A_j(z)e^{P_j(z)}+B_j(z)e^{-\lambda z^m}|\,|
\frac{f^{(j)}}{f}|.
\end{split}
\end{equation}
Since $\delta (\lambda z^m$, $\theta )>0$,
$\delta ( P_0,\theta )>0$ and $\delta (P_j,\theta )<0$
$(j=1,\dots ,k-1)$ for all $\theta \in (\theta _1,\theta
_2)$, by Lemma \ref{lem3}, for any $\theta \in (
\theta _1,\theta _2)$ there is $R_0(\theta )>0$
such that for $|z|=r>R_0$, we have
\begin{gather}
\exp \{ (1-\varepsilon )\delta (P_0,\theta )r^{n}\}
\leq |A_0(z)e^{P_0(z) }+B_0(z)e^{-\lambda z^m}|,  \label{f2}
\\
|A_j(z)e^{P_j(z)}+B_j(z)e^{-\lambda z^m}|
\leq C_1\quad (j=1,\dots ,k-1).  \label{f3}
\end{gather}
From Lemma \ref{lem1}, there exist a set $E\subset [0,2\pi )$
that has linear measure zero and a constant $M>0$ such that
 for any $ \theta \in [0,2\pi )\backslash E$ there exists a
constant $R_1=R_1(\theta )>1$ such that for all $z$ satisfying
$\arg z=\theta $ and $|z|=r>R_1$, we have
\begin{equation}
|\frac{f^{(j)}}{f}|\leq C_2[T(2r,f)] ^{2k}\quad (j=1,\dots ,k).
\label{f4}
\end{equation}
By using \eqref{f2}-\eqref{f4} in \eqref{f1}, for
$r>\max \{R_0,R_1\} $, we obtain
\[
\exp \{ (1-\varepsilon )\delta (P_0,\theta )
r^{n}\} \leq C_{3}[T(2r,f)] ^{2k},
\]
which implies that
$\sigma _2(f)\geq n$.
By Lemma \ref{lem2}, we obtain
$n\leq \sigma _2(f)\leq m$. %\label{f5}
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor1}]
In these conditions, there exist $\theta _1,\theta _2$ such
that $\theta _1<\theta _2$,  $\delta (\lambda z^m,\theta )>0$,
$\delta (P_0,\theta )>0$ and $\delta (P_j,\theta )<0$
$(j=1,\dots ,k-1)$
for all $\theta \in (\theta _1,\theta _2)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
Since $m>n$ and $a_{n,j}=c_ja_{n,0}$
$(0<c_j<1)$ $(j=1,\dots ,k-1)$,  there exist
$\theta _1<\theta _2$ such that $\delta (\lambda z^m,\theta )>0$
and $\delta (P_j,\theta )>0$ $(j=0,\dots ,k-1)$ for all
$\theta \in (\theta _1,\theta _2)$. In this case from
Lemma \ref{lem3}, for sufficiently large $r$, we have
\begin{gather}
\exp \{ (1-\varepsilon )\delta (P_0,\theta )r^{n}\}
\leq |A_0(z)e^{P_0(z)}+B_0(z)e^{-\lambda z^m}|,  \label{f6}
\\
|A_j(z)e^{P_j(z)}+B_j(z)e^{-\lambda z^m}|
\leq \exp \{ ( 1+\varepsilon )c\delta (P_0,\theta )r^{n}\} ,\label{f7}
\end{gather}
where $c=\max \{ c_j\}$. Using \eqref{f6}, \eqref{f7} and
\eqref{f4} in \eqref{f1}, for $r$ large enough,
\begin{equation}
\exp \{ (1-\varepsilon )\delta (P_0,\theta ) r^{n}\}
\leq C_{4}\exp \{ (1+\varepsilon )c\delta (P_0,\theta )r^{n}\}
[T(2r,f)]^{2k},  \label{f7b}
\end{equation}
and thus
\begin{equation}
\exp \{ (1-\varepsilon -(1+\varepsilon )c)
\delta (P_0,\theta )r^{n}\}
\leq C_{4}[T( 2r,f)] ^{2k}.  \label{f8}
\end{equation}
Taking $0<\varepsilon <\frac{1-c}{1+c}$, we obtain,
from \eqref{f8} and Lemma \ref{lem2}, the desired estimate
$n\leq \sigma _2(f)\leq m$.
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor2}]
In this case also there exist $\theta _1<\theta _2$ such that
$\delta (\lambda z^m,\theta )>0$,
$\delta (P_0,\theta ) >0$ and $\delta (P_{s},\theta )<0$
for all $\theta \in (\theta _1,\theta _2)$. We have
\begin{equation}
|A_{s}(z)e^{P_{s}(z)}+B_{s}(
z)e^{-\lambda z^m}|\leq C_5  \label{f8b}
\end{equation}
and also for $j$ such that $\arg a_{n,j}=\arg a_{n,s}$
\begin{equation}
|A_j(z)e^{P_j(z)}+B_j(z)e^{-\lambda z^m}|\leq C_{6}.  \label{f8c}
\end{equation}
Now for $j$ such that $a_{n,j}=c_ja_{n,0}$ $(0<c_j<1)$, we
have \eqref{f7}. By combining \eqref{f7}, \eqref{f8b}, \eqref{f8c}
with \eqref{f1} we obtain
$n\leq \sigma _2(f)$,
and by Lemma \ref{lem2} we obtain
$n\leq \sigma _2(f)\leq m$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
There exist $\theta _1<\theta _2$ such that $\delta (\lambda
z^m,\theta )>0$, $\delta (P_0-P_{s},\theta )>0$ and
$\delta (P_j-P_{s},\theta )>0$ for every
$\theta \in (\theta _1,\theta _2)$. Thus for $r$ sufficiently
large, we have
\begin{gather}
\exp \{ (1-\varepsilon )\delta (P_0-P_{s},\theta)r^{n}\}
 \leq |A_0(z)e^{P_0(z)-P_{s}(z)}+B_0(z)e^{-\lambda
z^m-P_{s}(z)}|,  \label{f9}
\\
|A_{s}(z)+B_{s}(z)e^{-\lambda z^m-P_{s}(z)}|
\leq \exp \{ r^{\sigma (A_{s})+\varepsilon }\} ,  \label{f10}
\\
|A_0(z)e^{P_j(z)-P_{s}(z)}+B_0(z)e^{-\lambda z^m-P_{s}(z)}|
\leq C_{7}.  \label{f10b}
\end{gather}
From \eqref{eq2}, we obtain
\begin{equation}
\begin{split}
&|A_0(z)e^{P_0(z)-P_{s}( z)}+B_0(z)e^{-\lambda z^m-P_{s}(z)}|\\
&\leq |e^{-\lambda z^m-P_{s}(z)}||\frac{f^{(k)}}{f}|
+\sum_{j=1}^{k-1}|A_j(z)e^{P_j( z)-P_{s}(z)}+B_j(z)e^{-\lambda
z^m-P_{s}(z)}||\frac{f^{(j)}}{f}|.
\end{split}\label{f11}
\end{equation}
Substituting \eqref{f9}-\eqref{f10b} and \eqref{f4} in
\eqref{f11}, we obtain
\[
\exp \{ (1-\varepsilon )\delta (Q-P,\theta )
r^{n}\} \leq C_{8}\exp \{ r^{\sigma (A_{s})
+\varepsilon }\} [T(2r,f)] ^{2k}.
\]
Which implies
$n\leq \sigma _2(f)$,
and by Lemma \ref{lem2}, we obtain
$n\leq \sigma _2(f)\leq m$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4}]
There exist $\theta _1<\theta _2$ such that
$\delta (\lambda z^m,\theta )>0$ and $\delta (P_j,\theta )>0$
$(j=0,\dots ,k-1)$ for all
$\theta \in (\theta _1,\theta _2)$. In this case from
Lemma \ref{lem3}, for sufficiently large $r$, we have
\begin{gather}
\exp \{ (1-\varepsilon )\delta (P_0-P_{s},\theta)r^{n}\}
\leq |A_0(z)e^{P_0(z)-P_{s}(z)}+B_0(z)e^{-\lambda
z^m-P_{s}(z)}|,  \label{f41}
\\
|A_{s}(z)+B_{s}(z)e^{-\lambda
z^m-P_{s}(z)}|\leq \exp \{ r^{\sigma (
A_{s})+\varepsilon }\}  \label{f42}
\end{gather}
and for $j\neq 0,s$
\begin{equation}
|A_j(z)e^{P_j(z)-P_{s}(z)}+B_j(z)e^{-\lambda z^m-P_{s}(z)}|
\leq \exp \{ (1+\varepsilon )c\delta (
P_0-P_{s},\theta )r^{n}\} ,  \label{f43}
\end{equation}
where $c=\max \{ c_j\} $. Using \eqref{f41}-\eqref{f43} and
\eqref{f4} in \eqref{f11}, for $r$ large enough, we obtain
%\begin{align*}
\[
\exp \{ (1-\varepsilon )\delta (P_0,\theta )r^{n}\}
\leq C_{9}\exp \{ r^{\sigma (A_{s})+\varepsilon }\} \exp
\{ (1+\varepsilon )c\delta (P_0,\theta )
r^{n}\} [T(2r,f)] ^{2k},
\] %\end{align*}
and thus
\begin{equation}
\exp \{ (1-\varepsilon -(1+\varepsilon )c) \delta (P_0,\theta )r^{n}\}
\leq C_{9}\exp \{r^{\sigma (A_{s})+\varepsilon }\} [T( 2r,f)] ^{2k}  \label{f44}
\end{equation}
By taking $0<\varepsilon <\frac{1-c}{1+c}$, from \eqref{f44} and
Lemma \ref{lem2}, we obtain
$n\leq \sigma _2(f)\leq m$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm5}]
By taking $B_j(z)\equiv 0$ $(j=0,\dots ,k-1)$ in
previous theorems, we obtain that every solution
$f(z) \not\equiv 0$ of \eqref{eq3} is of infinite order with
$n\leq \sigma_2(f)\leq m$. It remains to show that
 $\sigma _2(f)=m$ or $\sigma _2(f)=n$. For that we suppose the
contrary, i.e. $n<\sigma _2(f)<m$ and we prove that this
implies a contradiction.

Recall the Wiman-Valiron theory \cite{val}, there is a set
$E_1\subset [1,+\infty )$ that has finite logarithmic measure,
such that for $|z|=r\notin [0,1] \cup E_1$ and
$|f(z)|=M(r,f)$, we have
\begin{equation}
\frac{f^{(j)}(z)}{f(z)}=(\frac{
\nu (r)}{z})^{j}(1+o(1))\quad (j=1,\dots ,k-1),  \label{f111}
\end{equation}
where $\nu (r)$ is the central index of $f(z)$.

Set $\sigma _2(f)=\gamma $. From Lemma \ref{lem5}, we can take
a sequence of points $\{ z_{p}=r_{p}e^{i\theta _{p}}\} $ such
that $f(z_{p})=M(r_{p},f)$,
$\theta _{p}\in [0,2\pi )$,
$\lim_{p\to \infty }\theta _{p}=\theta _0\in [0,2\pi )$,
$r_{p}\notin [0,1] \cup E_1\cup E_2$, and for any given
$\epsilon >0$, as $r_{p}\to \infty $, we
have
\begin{equation}
\exp \{ r_{p}^{\gamma -\epsilon }\} \leq \nu (r_{p})
\leq \exp \{ r_{p}^{\gamma +\epsilon }\} .  \label{f112}
\end{equation}
From \eqref{eq3}, we can write
\begin{equation}
-\frac{f^{(k)}}{f}=\Big(A_{k-1}(z)
e^{P_{k-1}(z)}\frac{f^{(k-1)}}{f}+\dots +A_0(
z)e^{P_0(z)}\Big)e^{\lambda z^m}.  \label{f113}
\end{equation}
Using \eqref{f111} in \eqref{f113}, we obtain
\begin{equation} \label{f114}
\begin{split}
-\nu ^{k}(r_{p})(1+o(1))
&=(z_{p}A_{k-1}e^{P_{k-1}}\nu ^{k-1}(r_{p})(1+o(
1))+\dots \\
&\quad +z_{p}^{k-1}A_1e^{P_1}\nu (r_{p})(1+o(
1))+z_{p}^{k}A_0e^{Q})e^{\lambda z_{p}^m}.
\end{split}
\end{equation}
Now we prove three cases separately.

\textbf{Case 1.} $\delta (\lambda z^m,\theta _0)=:\delta
>0$. From \eqref{f112}, for $p$ sufficiently large, we obtain
\begin{equation}
|-\nu ^{k}(r_{p})(1+o(1))|\leq 2\exp \{ kr_{p}^{\gamma +\epsilon }\} .
\label{f115}
\end{equation}
From Lemma \ref{lem3} and by taking account $\gamma +\epsilon <m$,
 for $p$ large enough, we have
\begin{equation}  \label{f116}
\begin{split}
\exp \{ (1-\epsilon )\delta r_{p}^m\}
& \leq ( z_{p}A_{k-1}e^{P_{k-1}}\nu ^{k-1}(r_{p})(1+o(1))+\dots\\
&\quad +z_{p}^{k-1}A_1e^{P_1}\nu (r_{p})(1+o(
1))+z_{p}^{k}A_0e^{Q})e^{\lambda z_{p}^m}.
\end{split}
\end{equation}
By combining \eqref{f115} and \eqref{f116} with \eqref{f114}, a
contradiction follows.

\textbf{Case 2.} $\delta (\lambda z^m,\theta _0)=:\delta<0$.
 From \eqref{f112}, for $p$ large enough, we have
\begin{equation}
\frac{1}{2}\exp \{ kr_{p}^{\gamma -\epsilon }\} \leq |
-\nu ^{k}(r_{p})(1+o(1))|,
\label{f117}
\end{equation}
and from Lemma \ref{lem3}, we obtain
\begin{equation} \label{f118}
\begin{split}
&(z_{p}A_{k-1}e^{P_{k-1}}\nu ^{k-1}(r_{p})(1+o(1))+\dots\\
&+z_{p}^{k-1}A_1e^{P_1}\nu (r_{p})(1+o(
1))+z_{p}^{k}A_0e^{Q})e^{\lambda z_{p}^m}
\leq \exp \{ (1-\epsilon )\delta r_{p}^m\} .
\end{split}
\end{equation}
Also a contradiction follows from the combination
of \eqref{f117} and \eqref{f118} with \eqref{f114} as $p\to \infty $.

\textbf{Case 3.} $\delta (\lambda z^m,\theta _0)=0$.
Since $\lim_{p\to \infty }\theta _{p}=\theta _0$, then for $p$ large
enough, we obtain
\[
\frac{1}{e}\leq |e^{\lambda z_{p}^m}|\leq e.
\]
By Lemma \ref{lem3}, there exists $\alpha >0$ such that
\begin{align*}
&(z_{p}A_{k-1}e^{P_{k-1}}\nu ^{k-1}(r_{p})(1+o(1))+\dots
+z_{p}^{k-1}A_1e^{P_1}\nu (r_{p})(1+o(1))+z_{p}^{k}A_0e^{Q})
e^{\lambda z_{p}^m}\\
&\leq \exp \{\alpha r_{p}^{n}\} \nu ^{k-1}(r_{p}).
\end{align*}
Combining this with \eqref{f114}, we obtain
\[
\frac{1}{2}|\nu (r_{p})|\leq \exp \{\alpha r_{p}^{n}\} ;
\]
and with
\[
\exp \{ r_{p}^{\gamma -\epsilon }\} \leq |\nu (
r_{p})|,
\]
and by taking account $n<\gamma -\epsilon $, provided $\epsilon $
small enough, a contradiction follows.
\end{proof}

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\end{document}