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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 108, 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  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/108\hfil Small functions and the solutions]
{Relations between the small functions and the solutions of
certain second-order differential equations}

\author[H. Liu, Z. Mao \hfil EJDE-2007/108\hfilneg]
{Huifang Liu, Zhiqiang Mao}  % in alphabetical order

\address{Huifang Liu \newline
School of Mathematics, South China Normal University, Guangzhou
510631, China. \newline
Institute of Mathematics and Informatics,
Jiangxi Normal University, Nanchang 330027, China}
\email{liuhuifang73@sina.com}

\address{Zhiqiang Mao\newline
School of Mathematics, South China Normal University,
Guangzhou 510631, China. \newline
Depatment of Mathematics, Jiangxi Science and Teachers college,
Nanchang 330013, China}
\email{maozhiqiang1@sina.com}

\thanks{Submitted February 12, 2007. Published August 7, 2007.}
\thanks{Supported by grant 04010360 from the Natural Science Foundation
 of Guangdong, China}
\subjclass[2000]{34M10}
\keywords{Entire function; exponent of convergence of the zero-sequence}

\begin{abstract}
 In this paper, we investigate the relations between the small
 functions and the solutions, first, second derivatives, and differential
 polynomial of the solutions to the differential equation
 $$
 f''+A_1e^{P(z)}f'+A_0e^{Q(z)}f=0,
 $$
 where $P(z)=a_nz^n+\dots+a_0$, $Q(z)=b_nz^n+\dots+b_0$  are polynomials
 with degree $n$ ($n\geq1$),
 $a_i$, $b_i$ ($i=0,1,\dots,n$), $a_nb_n\neq 0$ are complex constants,
 $A_j(z)\not \equiv 0$ ($j=0,1$) are entire functions with $\sigma(A_j)<n$.
\end{abstract}

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

\section{Main Results}

In this paper, we use the standard notation of Nevanlinna's
value distribution theory \cite{h1}. In addition, we  use
notations $\sigma(f)$, $\lambda(f)$, $\overline{\lambda}(f)$ to
denote the order of growth, the exponent of convergence of
the zero-sequence and the sequence of distinct zeros of $f(z)$ respectively.
A meromorphic function $g(z)$ is called a small function of a
meromorphic function $f(z)$ if $T(r, g)=o(T(r, f))$, as
$r\to +\infty$.

 Consider the differential equation
\begin{equation}
f''+A_1e^{P(z)}f'+A_0e^{Q(z)}f=0, \label{e1.1}
\end{equation}
where $P(z)$, $Q(z)$ are polynomials with degree $n$ ($n\geq1$),
 $A_j(z)\not \equiv 0$ ($j=0,1$) are entire functions with
$\sigma(A_1)<\deg P$, $\sigma(A_0)<\deg Q$.
 If $\deg P\neq\deg Q$, then every solution of \eqref{e1.1} has infinite
order \cite[p. 419]{g1}.
 If $\deg P=\deg Q$, then equation \eqref{e1.1} may have a solution of finite
 order. Indeed  $f(z)=z$ satisfies
$f''+ze^zf'-e^zf=0$.   Kwon \cite{k1}  studied the growth of
solutions of equation \eqref{e1.1} with $\deg P=\deg Q$, and obtained
the following result.

\begin{theorem} \label{thmA}
Let $A_j(z)\not\equiv 0$ ($j=0,1$) be entire
functions with $\sigma(A_j)<n$, $P(z)=a_nz^n+\dots+a_0$,
$Q(z)=b_nz^n+\dots+b_0$ be polynomials with degree $n$ ($n\geq1$),
where  $a_i$, $b_i$ ($i=0,1,\dots,n$), $a_nb_n\neq 0$ are complex
constants such that $\arg{a_n}\neq\arg{b_n}$ or $a_n=cb_n(0<c<1)$.
 Then every solution $f\not\equiv 0$ of equation \eqref{e1.1}
has infinite order.
\end{theorem}

Chen and  Shon \cite{c4} studied the differential equation
\begin{equation}
f''+A_1e^{az}f'+A_0e^{bz}f=0 \label{e1.2}
\end{equation}
and obtained

\begin{theorem} \label{thmB}
 Let $A_j(z)\not\equiv 0$ ($j=0,1$) be entire
functions with $\sigma(A_j)<1$, $a$, $b$ be complex constants such
that $ab\neq 0$ and $\arg{a}\neq\arg{b}$ or $a=cb(0<c<1)$.  If
$\varphi(z)\not\equiv 0$ is an entire function with finite order,
then every solution $f\not\equiv 0$ of equation \eqref{e1.2} satisfies
$\overline{\lambda}(f-\varphi)=\overline{\lambda}(f'-\varphi)
=\overline{\lambda}(f''-\varphi)=\infty$.
\end{theorem}

\begin{theorem} \label{thmC}
 Let $A_j(z),a,b$  satisfy  the  hypotheses  of
 Theorem  \ref{thmB}, $d_0(z)$, $d_1(z)$, $d_2(z)$ be polynomials  not
 all equal to zero,
$\varphi(z)\not\equiv 0$ is an entire function of
 order  less than 1. If $f\not\equiv 0$ is a solution of equation
\eqref{e1.2}, then the differential
polynomials $g(z)=d_2f''+d_1f'+d_0f$ satisfy
$\overline{\lambda}(g-\varphi)=\infty$.
\end{theorem}

In this paper we go deeply into the study of the relations of the
small functions and solutions
   of the differential equation \eqref{e1.1} and obtain the following
theorem.

 \begin{theorem} \label{thm1}
Let  $A_j(z)\not\equiv 0$ ($j=0,1$), $P(z)$,
$Q(z)$ satisfy the hypotheses of Theorem \ref{thmA}. If
$\varphi(z)\not\equiv 0$ is an entire function with finite order,
then every solution $f \not\equiv 0$ of equation \eqref{e1.1} satisfies
$\overline{\lambda}(f-\varphi)=\overline{\lambda}(f'-\varphi)
=\overline{\lambda}(f''-\varphi)=\infty$.
\end{theorem}

\begin{theorem} \label{thm2}
 Let  $A_j(z)\not\equiv 0)$ ($j=0,1$), $P(z)$, $Q(z)$
satisfy the hypotheses of Theorem \ref{thmA}, $d_0(z)$, $d_1(z)$, $d_2(z)$
be polynomials that are not all equal to zero,
$\varphi(z)\not\equiv 0$ is an entire function of order that is
less than $n$ . If $f \not\equiv 0$ is a solution of equation
\eqref{e1.1}, then  the differential polynomials
$g(z)=d_2f''+d_1f'+d_0f$ satisfy $\overline{\lambda}(g-\varphi)=\infty$.
\end{theorem}

\section{Auxiliary Lemmas}


\begin{lemma}[\cite{c3}] \label{lem1}
 Let $f(z)$ be a transcendental meromorphic
function with $\sigma(f)=\sigma<+\infty$. Then for any given
$\varepsilon>0$, there is a set $E\subset[0,2\pi)$ that has linear
measure zero, such that if $\varphi\in[0,2\pi)\setminus E$, then
there is a constant $R_0=R_0(\varphi)>1$, such that for all $z$
satisfying $\arg z=\varphi$ and $|z|\geq R_0$, we have
$\exp{\{-r^{\sigma+\varepsilon}\}}\leq|f(z)|
\leq\exp{\{r^{\sigma+\varepsilon}\}}$.
\end{lemma}


\begin{lemma} \label{lem2}
Let $P(z)=(\alpha+i\beta)z^n+\dots$ ($\alpha,\beta$
are real, $|\alpha|+|\beta|\neq0)$ be a polynomial with degree
$n\geq1$, $A(z)\not \equiv 0$ be a meromorphic 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 $\varepsilon>0$, there is a set $H_1\subset[0,2\pi)$ that
has linear measure zero, such that for any
$\theta\in[0,2\pi)\setminus(H_1\bigcup H_2)$  and a sufficiently
large $r$, we have
\begin{itemize}
\item[(i)] If $\delta(P,\theta)>0$, then
\[
\exp{\{(1-\varepsilon)\delta(P,\theta)r^n\}}\leq|g(re^{i\theta})|
\leq\exp{\{(1+\varepsilon)\delta(P,\theta)r^n\}};
\]

\item[(ii)] If $\delta(P,\theta)<0$, then
\[
\exp{\{(1+\varepsilon)\delta(P,\theta)r^n\}}\leq|g(re^{i\theta})|
\leq\exp{\{(1-\varepsilon)\delta(P,\theta)r^n\}},
\]
where $H_2=\{\theta\in[0,2\pi);\delta(P,\theta)=0\}$ is a finite
set.
\end{itemize}
\end{lemma}

\begin{proof} Rewrite $g(z)$ as
$g(z)=we^{(\alpha+i\beta)z^n}$, where
$w(z)=A(z)e^{P(z)-(\alpha+i\beta)z^n}$ is a meromorphic function
with $\sigma(w)=s<n$. By lemma \ref{lem1}, for any given $\varepsilon
(0<\varepsilon<n-s)$, there is a set $H_1\subset[0,2\pi)$ that has
linear measure zero, such that, if $\theta\in[0,2\pi)\setminus H_1$,
then there exists a constant $R=R(\theta)>1$, for all $z$ satisfying
$\arg z=\theta$ and $|z|\geq R$, we have
$\exp{\{-r^{s+\varepsilon}\}}\leq|w(z)|\leq\exp{\{r^{s+\varepsilon}\}}$.
By
$|e^{(\alpha+i\beta)z^n}|=e^{{Re}(\alpha+i\beta)z^n=e^{\delta(P,\theta)r^n}}$,
when $\theta\in[0,2\pi)\setminus(H_1\bigcup H_2)$ and  $|z|=r>R$, we
have
$\exp{\{-r^{s+\varepsilon}+\delta(P,\theta)r^n\}}\leq|g(z)|\leq\exp{\{r^{s+\varepsilon}+\delta(P,\theta)r^n\}}$.
So by the above inequality and $\delta(P,\theta)>0$ or
$\delta(P,\theta)<0$, we complete the proof.
\end{proof}

\begin{lemma}[\cite{c4}] \label{lem3}
  Let $f(z)$ be an entire function with
infinite order, $d_j(z)$
($j=0,1,2$) be polynomials that are not all
equal to zero. Then
\[
w(z)=d_2f''+d_1f'+d_0f
\]
has infinite order.
\end{lemma}

\begin{lemma} \label{lem4}
  Let $a_i,b_i(i=0,1\dots n)$
 be complex constants such that $a_nb_n\neq 0$ and $\arg{a_n}\neq\arg{b_n}$
or $a_n=cb_n (0<c<1)$,
$P(z)=a_nz^n+\dots+a_0$, $Q(z)=b_nz^n+\dots+b_0$. We denote index
sets by
\begin{gather*}
\Lambda_1=\{0,P\};\\
\Lambda_2=\{0,P,Q,2P,P+Q\}; \\
\Lambda_3=\{0,P,Q,2P,P+Q,2Q,3P,2P+Q,P+2Q\}; \\
\Lambda_4=\big\{0,P,Q,2P,P+Q,2Q,3P,2P+Q,P+2Q,\\
3Q, 4P,3P+Q,2P+2Q,P+3Q\big\}.
\end{gather*}
Then
\begin{itemize}
\item[(i)]  If $H_j$ ($j \in \Lambda_1$) and $H_Q$ are all
meromorphic functions of orders that are less than
 $n$, $H_Q \not\equiv 0$, setting $\psi_1(z)=\Sigma_{j \in
\Lambda_1}H_j(z)e^{j}$, then $\psi_1(z)+H_Qe^{Q}\not\equiv 0$.

\item[(ii)]  If $H_j$ ($j \in \Lambda_2$) and $H_{2Q}$ are all
meromorphic functions of orders that are less than
 $n$, $H_{2Q} \not\equiv 0$, setting $\psi_2(z)=\Sigma_{j \in
\Lambda_2}H_j(z)e^{j}$, then $\psi_2(z)+H_{2Q}e^{{2Q}}\not\equiv 0$.

\item[(iii)]  If $H_j(j \in \Lambda_3)$ and $H_{3Q}$ are all
meromorphic functions of orders that are less than
 $n$, $H_{3Q} \not\equiv 0$, setting $\psi_3(z)=\Sigma_{j \in
\Lambda_3}H_j(z)e^{j}$, then $\psi_3(z)+H_{3Q}e^{{3Q}}\not\equiv 0$.

\item[(iv)]  If $H_j(j \in \Lambda_4)$ and $H_{4Q}$ are all
meromorphic functions of orders that are less than
 $n$, $H_{4Q} \not\equiv 0$, setting $\psi_4(z)=\Sigma_{j \in
\Lambda_4}H_j(z)e^{j}$, then $\psi_4(z)+H_{4Q}e^{{4Q}}\not\equiv 0$.

\item[(v)] The derived function of $\psi_j(z)$ ($j=1,\dots,4$)
keep the above properties of
 $\psi_j(z)$, and also it can be expressed by  $\psi_j(z)$.
$\psi_j(z)$ may be different at different places, but preserve the
above properties. $\psi_2(z)\psi_2(z)$( it denotes the product of
two $\psi_2(z)$, and two $\psi_2(z)$ may be different.) is of
properties of
 $\psi_4(z)$, we write $\psi_2(z)\psi_2(z)=\psi_4(z)$. Similarly we
 have
\[
\psi_1(z)\psi_1(z)=\psi_2(z),\psi_1(z)\psi_2(z)
=\psi_3(z),\psi_1(z)\psi_3(z)=\psi_4(z).
\]


\item[(vi)]  let $\psi_{20}(z),\psi_{21}(z),\psi_{22}(z)$ have
the form of $\psi_2(z)$ which is defined as in (ii), $\varphi(z)\not
\equiv 0$ is a meromorphic function with finite order and $H_{2Q}
\not\equiv 0$ are all meromorphic functions of orders that are less
than  $n$.
Then
\[
\frac{\varphi''(z)}{\varphi(z)}\psi_{22}(z)
+\frac{\varphi'(z)}{\varphi(z)}\psi_{21}(z)
+\psi_{20}(z)+H_{2Q}e^{{2Q}}\not\equiv 0.
\]
\end{itemize}
\end{lemma}

\begin{proof} Properties (i)--(iv) are
similar, and the properties of $\psi_j(z)$ ($j=1,\dots,4$) in (v) are
clear, so we only prove (ii) and  (vi).
For the proof of (ii). We consider two cases:

\noindent\textbf{Case 1: $\arg{a_n}\neq\arg{b_n}$.}
 Then $\arg({a_n+b_n})$, $\arg{a_n}$, $\arg{b_n}$ are three distinct
arguments. Set $\sigma(H_0)=\beta<n$, by Lemma \ref{lem1},
for any given $\varepsilon(0<\varepsilon<\min\{\frac{1}{5},n-\beta\})$,
 there exists a set  $E_0\subset[0,2\pi)$ that has linear measure zero,
such that if $\theta\in[0,2\pi)\setminus E_0$, then there is a
constant $R=R(\theta)>1$, such that
for all $z$ satisfying  $\arg z=\theta$ and $|z|=r\geq R$, we have
\begin{equation}
|H_0(z)|\leq\exp{\{r^{\beta+\varepsilon}\}}. \label{e2.1}
\end{equation}
By lemma \ref{lem2}, there exists a ray
$\arg z=\theta\in[0,2\pi) \setminus
(E_0\cup E_1\cup E_2)$, where $E_1\subset[0,2\pi)$ has linear
measure zero, $E_2=\{\theta\in[0,2\pi);\delta(P,\theta)=0$ or
$\delta(Q,\theta)=0$ or $\delta(P+Q,\theta)=0\}$  is a finite set,
such that
$\delta(P,\theta)<0$, $\delta(P+Q,\theta)<0$,
$\delta(Q,\theta)>0$, and
for the above given $\varepsilon$, we have, when $r$ is sufficiently
large,
\begin{gather}
|H_{2Q}e^{2Q}|\geq\exp{\{(1-\varepsilon)2\delta(Q,\theta)r^n\}}, \label{e2.2}
\\
|H_{Q}e^{Q}|\leq\exp{\{(1+\varepsilon)\delta(Q,\theta)r^n\}}, \label{e2.3}
\\
|H_{P+Q}e^{P+Q}|\leq\exp{\{(1-\varepsilon)\delta(P+Q,\theta)r^n\}}<1,
 \label{e2.4}\\
|H_{2P}e^{2P}|\leq\exp{\{(1-\varepsilon)2\delta(P,\theta)r^n\}}<1, \label{e2.5}
\\
|H_{P}e^{P}|\leq\exp{\{(1-\varepsilon)\delta(P,\theta)r^n\}}<1. \label{e2.6}
\end{gather}
If $\psi_2(z)+H_{2Q}e^{{2Q}}\equiv 0$, then by \eqref{e2.1}-\eqref{e2.6},
we have
\begin{align*}
\exp{\{(1-\varepsilon)2\delta(Q,\theta)r^n\}}
&\leq|H_{2Q}e^{2Q}|\\
&\leq\exp{\{r^{\beta+\varepsilon}\}}+\exp{\{(1+\varepsilon)
 \delta(Q,\theta)r^n\}}+3\\
&\leq 3\exp{\{r^{\beta+\varepsilon}\}}\exp{\{(1+\varepsilon)
 \delta(Q,\theta)r^n\}}.
\end{align*}
Because $2(1-\varepsilon)-(1+\varepsilon)=1-3\varepsilon>\frac{2}{5}$, we
have
\[
\exp{\{\frac{2}{5}\delta(Q,\theta)r^n\}}\leq3\exp{\{r^{\beta+\varepsilon}\}}.
\]
 This is a contradiction to $\beta+\varepsilon<n$.
Hence $\psi_2(z)+H_{2Q}e^{{2Q}}\not\equiv 0$.
\smallskip

\noindent\textbf{Case 2: $a_n=cb_n(0<c<1)$.}
 Set $\sigma(H_0)=\beta<n$. By Lemmas \ref{lem1} and \ref{lem2}, for any given
 $\varepsilon(0<\varepsilon<\min\{\frac{1-c}{5},n-\beta\})$,
there exist set $E_j\subset[0,2\pi)(j=0,1,2)$ that have linear
measure zero, $E_j$ are defined as in the case (1) respectively. We
take the ray  $\theta\in[0,2\pi) \setminus (E_0\cup E_1\cup E_2)$,
such that $\delta(Q,\theta)>0$, and when $|z|=r$ is sufficiently
large, we have \eqref{e2.1}-\eqref{e2.3} and
\begin{gather}
|H_{P+Q}e^{P+Q}|\leq\exp{\{(1+\varepsilon)(1+c)\delta(Q,\theta)r^n\}},
  \label{e2.7}\\
|H_{2P}e^{2P}|\leq\exp{\{(1+\varepsilon)2c\delta(Q,\theta)r^n\}},
  \label{e2.8}\\
|H_{P}e^{P}|\leq\exp{\{(1+\varepsilon)c\delta(Q,\theta)r^n\}}. \label{e2.9}
\end{gather}
If $\psi_2(z)+H_{2Q}e^{{2Q}}\equiv 0$, then by \eqref{e2.1}-\eqref{e2.3},
and \eqref{e2.7}-\eqref{e2.9}, we have
\begin{equation}
\begin{aligned}
\exp{\{(1-\varepsilon)2\delta(Q,\theta)r^n\}}
&\leq|H_{2Q}e^{2Q}|\\
&\leq\exp{\{r^{\beta+\varepsilon}\}}
  +2\exp{\{(1+\varepsilon)(1+c)\delta(Q,\theta)r^n\}}\\
&\quad+ \exp{\{(1+\varepsilon)2c\delta(Q,\theta)r^n\}}
  +\exp{\{(1+\varepsilon)c\delta(Q,\theta)r^n\}}.
\end{aligned}  \label{e2.10}
\end{equation}
Because $0<\varepsilon<\min\{\frac{1-c}{5},n-\beta\}$,
 when $r\to+\infty$, we have
\begin{gather}
\frac{\exp{\{r^{\beta+\varepsilon}\}}}{\exp{\{(1-\varepsilon)
2\delta(Q,\theta)r^n\}}}\to 0, \label{e2.11}\\[5pt]
\frac{\exp{\{(1+\varepsilon)(1+c)\delta(Q,\theta)r^n\}}}{\exp{\{(1-\varepsilon)
 2\delta(Q,\theta)r^n\}}}\to 0, \label{e2.12}\\[5pt]
\frac{\exp{\{(1+\varepsilon)2c\delta(Q,\theta)r^n\}}}{\exp{\{(1-\varepsilon)
 2\delta(Q,\theta)r^n\}}}\to 0, \label{e2.13}\\[5pt]
\frac{\exp{\{(1+\varepsilon)c\delta(Q,\theta)r^n\}}}{\exp{\{(1-\varepsilon)
 2\delta(Q,\theta)r^n\}}}\to 0. \label{e2.14}
\end{gather}
By \eqref{e2.10}-\eqref{e2.14}, we get a contradiction. Hence
$\psi_2(z)+H_{2Q}e^{{2Q}}\not\equiv 0$.
\smallskip

Proof of (vi).  By $\sigma(\varphi)<\infty$ and \cite[p. 89]{g2}
we know, for any given  $\varepsilon>0$, there exists a set
$E\subset[0,2\pi)$ that has linear measure zero,
if $\theta\in[0,2\pi)\setminus E$, then there exists a constant
 $R=R(\theta)>1$, such that for all $z$ satisfying
$\arg z=\theta$ and $|z|\geq R$, we have
\[
|\frac{\varphi^{(k)}(z)}{\varphi(z)}|\leq|z|^{k(\sigma(\varphi)-1
+\varepsilon)}\quad  (k=1,2).
\]
So on the ray $\arg z=\theta\in[0,2\pi)\setminus E$,
$\frac{\varphi^{(k)}(z)}{\varphi(z)}H_j(z)e^j$
($k=1,2,j\in\Lambda_2$) keep the properties of $H_je^j$ which are
defined as
 in \eqref{e2.1}, \eqref{e2.3}--\eqref{e2.6} or \eqref{e2.1}, \eqref{e2.3},
\eqref{e2.7}--\eqref{e2.9}.
Using a similar reasoning to that in the proof of (ii), we can prove
(vi).
\end{proof}



\begin{lemma}[\cite{c2}] \label{lem5}
Suppose that $A_0,\dots,A_{k-1},F\not\equiv 0$ are finite-order meromorphic
functions. If $f$ is an infinite-order meromorphic solution of the equation
\[
f^{(k)}+A_{k-1}f^{(k-1)}+\dots+A_0f=F,
\]
then $f$ satisfies $\overline{\lambda}(f)=\lambda(f)=\sigma(f)=\infty$.
\end{lemma}

\section{Proofs of Theorems}

\begin{proof}[Proof of Theorem \ref{thm1}]
 Suppose that $f(z)\not \equiv 0$ is a
solution of \eqref{e1.1}. First of all we prove that
$\overline{\lambda}(f-\varphi)=\infty$.
Set
$g_0=f-\varphi$, then
$\sigma(g_0)=\sigma(f)=\infty$,$\overline{\lambda}(g_0)=\overline{\lambda}(f-\varphi)$.
Substituting
$f=g_0+\varphi$,$f'=g'_0+\varphi'$,$f''=g''_0+\varphi''$ into
equation  \eqref{e1.1}, we have
\begin{equation}
g''_0+A_1e^{P(z)}g'_0+A_0e^{Q(z)}g_0=-(\varphi''+A_1e^{P(z)}\varphi'
+A_0e^{Q(z)}\varphi). \label{e3.1}
\end{equation}
We remark that \eqref{e3.1} may have finite-order solution (For
example  when $\varphi(z)=z$, $g_0=-z$ solves the equation \eqref{e3.1}).
But here we  discuss only the case  $\sigma(g_0)=\infty$.

By $\varphi(z)$  being a finite-order entire function and Theorem \ref{thmA},
 we know  $\varphi''+A_1e^{P(z)}\varphi'+A_0e^{Q(z)}\varphi\not\equiv0$.
Hence by lemma \ref{lem5}, we have
$\overline{\lambda}(g_0)=\sigma(g_0)=\infty$, i.e.
$\overline{\lambda}(f-\varphi)=\infty$.

Secondly we prove $\overline{\lambda}(f'-\varphi)=\infty$.
Set $g_1=f'-\varphi$, then
$\sigma(g_1)=\sigma(f')=\sigma(f)=\infty$,$\overline{\lambda}(g_1)=\overline{\lambda}(f'-\varphi)$.
Differentiating both sides of equation \eqref{e1.1}, we get
\begin{equation}
f'''+A_1e^{P(z)}f''+[(A_1e^{P(z)})'+A_0e^{Q(z)}]f'+(A_0e^{Q(z)})'f=0. \label{e3.2}
\end{equation}
Substituting $f=-\frac{1}{A_0e^{Q(z)}}[f''+A_1e^{P(z)}f']$ into
 \eqref{e3.2}, we get
\begin{equation}
f'''+[A_1e^{P(z)}-\frac{(A_0e^{Q(z)})'}{A_0e^{Q(z)}}]f''
+[(A_1e^{P(z)})'+A_0e^{Q(z)}-\frac{(A_0e^{Q(z)})'}{A_0e^{Q(z)}}A_1e^{P(z)}]f'=0. \label{e3.3}
\end{equation}
 Substituting $f'=g_1+\varphi$, $f''=g'_1+\varphi'$, $f'''=g''_1+\varphi''$
into equation \eqref{e3.3},  we get
\begin{equation}
g''_1+h_1g'_1+h_0g_1=h, \label{e3.4}
\end{equation}
where $h_1=A_1e^{P(z)}-\frac{(A_0e^{Q(z)})'}{A_0e^{Q(z)}}$,
\begin{gather*}
h_0=(A_1e^{P(z)})'+A_0e^{Q(z)}-\frac{(A_0e^{Q(z)})'}{A_0e^{Q(z)}}A_1e^{P(z)},\\
-h=\varphi''-(\frac{A'_0}{A_0}+Q')\varphi'+[A_1\varphi'+A'_1\varphi
+P'A_1\varphi-\frac{A'_0}{A_0}A_1\varphi-Q'A_1\varphi]e^P+A_0\varphi e^Q.
\end{gather*}
Now we prove  $h\not\equiv0$. If  $h\equiv0$, then
\begin{equation}
\frac{\varphi''}{\varphi}-(\frac{A'_0}{A_0}+Q')\frac{\varphi'}{\varphi}+
[\frac{\varphi'}{\varphi}+\frac{A'_1}{A_1}+P'-\frac{A'_0}{A_0}-Q']A_1e^P+A_0
e^Q=0. \label{e3.5}
\end{equation}
By $\sigma(\varphi)<\infty$, $\sigma(A_j)<n$
($j=0,1$) and \cite[p. 89]{g2}, for any given
 $0<\varepsilon<\frac{1-c}{1+2c}$ ($c$ is defined as in Theorem \ref{thm1}),
 there exists a set $E_0\subset[0,2\pi)$ that has linear measure zero,
if $\theta\in[0,2\pi)\setminus E_0$, then there exists a constant
 $R=R(\theta)>1$, such that for all $z$ satisfying $\arg z=\theta$ and
$|z|\geq R$, we have
\begin{gather}
|\frac{\varphi^{(k)}(z)}{\varphi(z)}|\leq|z|^{k(\sigma(\varphi)-1
+\varepsilon)}\quad (k=1,2), \label{e3.6} \\
|\frac{A'_j(z)}{A_j(z)}|\leq|z|^{\sigma(A_j)-1+\varepsilon}\quad
 (j=0,1). \label{e3.7}
\end{gather}
Since $P(z)$, $Q(z)$ are polynomials with degree $n$,  when
$|z|=r$ is sufficiently large, we have
\begin{equation}
|P'(z)|\leq r^n\quad  \text{and}\quad  |Q'(z)|\leq r^n . \label{e3.8}
\end{equation}
So by \eqref{e3.6}-\eqref{e3.8}, there exists a positive constant $M$, such that
for all $z$ satisfying $\arg z=\theta\in[0,2\pi)\setminus E_0$, we
have, when $|z|=r$ is sufficiently large,
\begin{gather}
\big|(\frac{A'_0}{A_0}+Q')\frac{\varphi'}{\varphi}\big|\leq r^M, \label{e3.9} \\
\big|\frac{\varphi'}{\varphi}+\frac{A'_1}{A_1}+P'-\frac{A'_0}{A_0}-Q'\big|\leq r^M.
\label{e3.10}
\end{gather}
If $\arg{a_n}\neq\arg{b_n}$, then by lemma \ref{lem2}, there exists a ray
$\arg z=\theta\in[0,2\pi) \setminus (E_0\cup E_1\cup E_2)$,
$E_1\subset[0,2\pi)$ having linear measure zero,
$E_2=\{\theta\in[0,2\pi);\delta(P,\theta)=0$ or
$\delta(Q,\theta)=0\}$ being a finite set, such that
$\delta(P,\theta)<0, \delta(Q,\theta)>0$, and for the above given
$\varepsilon$, we have, when $r$ is sufficiently large,
\begin{gather}
|A_0e^{Q}|\geq\exp{\{(1-\varepsilon)\delta(Q,\theta)r^n\}}, \label{e3.11} \\
|A_1e^{P}|\leq\exp{\{(1-\varepsilon)\delta(P,\theta)r^n\}}<1. \label{e3.12}
\end{gather}
So by \eqref{e3.5},\eqref{e3.6} and \eqref{e3.9}-\eqref{e3.12}, we get
\[
\exp{\{(1-\varepsilon)\delta(Q,\theta)r^n\}}\leq|A_0e^{Q}|
\leq r^{2(\sigma(\varphi)-1+\varepsilon)}+r^M+r^M.
\]
This is absurd.

If $a_n=cb_n(0<c<1)$, then by lemma \ref{lem2}, there exists a ray $\arg
z=\theta\in[0,2\pi) \setminus (E_0\cup E_1\cup E_2)$, where $E_0,
E_1$ and $E_2$ are defined as the above, such that
$\delta(Q,\theta)>0$, and for the above given $\varepsilon$, when
$r$ is sufficiently large, we have \eqref{e3.11} and
\begin{equation}
|A_1e^{P}|\leq\exp{\{(1+\varepsilon)c\delta(Q,\theta)r^n\}}. \label{e3.13}
\end{equation}
So by \eqref{e3.5}, \eqref{e3.6}, \eqref{e3.9}-\eqref{e3.11} and \eqref{e3.13},
we get
\begin{align*}
\exp{\{(1-\varepsilon)\delta(Q,\theta)r^n\}}&\leq|A_0e^{Q}|\\
&\leq r^{2(\sigma(\varphi)-1+\varepsilon)}+r^M
 +r^M\exp{\{(1+\varepsilon)c\delta(Q,\theta)r^n\}}\\
&\leq3\exp{\{(1+2\varepsilon)c\delta(Q,\theta)r^n\}}.
\end{align*}
This is a contradiction to $0<\varepsilon<\frac{1-c}{1+2c}$. From
the above proof, we get $h\not\equiv0$.
 From $h\not\equiv0$ and lemma \ref{lem5} we get
 $\overline{\lambda}(g_1)=\sigma(g_1)=\infty$.
Hence $\overline{\lambda}(f'-\varphi)=\infty$.

Finally we prove that  $\overline{\lambda}(f''-\varphi)=\infty$.
Set
 $g_2=f''-\varphi$, then $\sigma(g_2)=\sigma(f'')=\sigma(f)=\infty$,
$\overline{\lambda}(g_2)=\overline{\lambda}(f''-\varphi)$.
Differentiating both sides of equation \eqref{e3.2}, we get
\begin{equation}
f^{(4)}+A_1e^{P}f'''+[2(A_1e^{P})'+A_0e^{Q}]f''+[(A_1e^{P})''+2(A_0e^{Q})']f'
+(A_0e^{Q})''f=0. \label{e3.14}
\end{equation}
Substituting $f=-\frac{1}{A_0e^{Q}}[f''+A_1e^{P}f']$ into \eqref{e3.14}, we
get
\begin{equation}
\begin{aligned}
f^{(4)}+A_1e^{P}f'''+[2(A_1e^{P})'+A_0e^{Q}-\frac{(A_0e^{Q})''}{A_0e^{Q}}]f''&\\
+[(A_1e^{P})''+2(A_0e^{Q})'-\frac{(A_0e^{Q})''}{A_0e^{Q}}A_1e^{P}]f'&=0.
\end{aligned}  \label{e3.15}
\end{equation}
By \eqref{e3.3} and  \eqref{e3.15}, we have
\begin{equation}
f^{(4)}+H_3f'''+H_2f''=0, \label{e3.16}
\end{equation}
where
\begin{gather}
H_3=A_1e^{P}-\frac{\varphi_1(z)}{\varphi_2(z)}, \label{e3.17} \\
H_2=2(A_1e^{P})'+A_0e^{Q}-\frac{(A_0e^{Q})''}{A_0e^{Q}}
  -\frac{\varphi_1(z)}{\varphi_2(z)}[A_1e^{P}-\frac{(A_0e^{Q})'}{A_0e^{Q}}],
 \label{e3.18}\\
\varphi_1(z)=(A_1e^{P})''+2(A_0e^{Q})'-\frac{(A_0e^{Q})''}{A_0e^{Q}}A_1e^{P},
 \label{e3.19}\\
\varphi_2(z)=(A_1e^{P})'+A_0e^{Q}-\frac{(A_0e^{Q})'}{A_0e^{Q}}A_1e^{P},
  \label{e3.20}
\end{gather}
and $\varphi_2(z) \not \equiv 0$ by Lemma \ref{lem4} (i). Clearly,
$H_3,H_2,\varphi_1(z),\varphi_2(z)$ are meromorphic functions with
$\sigma(\varphi_k)\leq n(k=1,2)$, $\sigma(H_j)\leq n(j=2,3)$.

Substituting
$f''=g_2+\varphi$,$f'''=g'_2+\varphi'$,$f^{(4)}=g''_2+\varphi''$
into  \eqref{e3.16},
\[
g''_2+H_3g'_2+H_2g_2=-(\varphi''+H_3\varphi'+H_2\varphi).
\]
If we can prove that $-(\varphi''+H_3\varphi'+H_2\varphi)\not\equiv0$,
then by lemma \ref{lem5}, we get $\overline{\lambda}(g_2)=\sigma(g_2)=\infty$.
Hence $\overline{\lambda}(f''-\varphi)=\infty$.
Now we prove $-(\varphi''+H_3\varphi'+H_2\varphi)\not\equiv0$.
Notice that
\begin{gather*}
(A_1e^P)'=(A_1'+A_1P')e^P,\quad
(A_1e^P)''=(A_1''+2A_1'P'+A_1(P')^2+A_1P'')e^P,\\
\frac{(A_0e^Q)'}{A_0e^Q}=\frac{A_0'}{A_0}+Q',\quad
\frac{(A_0e^Q)''}{A_0e^Q}=\frac{A_0''}{A_0}+2\frac{A_0'}{A_0}Q'+(Q')^2+Q''.
\end{gather*}
So by \eqref{e3.17}-\eqref{e3.20}, we have
\begin{gather}
\varphi_1(z)=B_1e^P+2(A_0'+A_0Q')e^Q, \label{e3.21} \\
\varphi_2(z)=B_2e^P+A_0e^Q, \label{e3.22}\\
H_3=\frac{1}{\varphi_2(z)}H_4, \label{e3.23}\\
H_2=\frac{1}{\varphi_2(z)}[A_0^2e^{2Q}+H_5], \label{e3.24}
\end{gather}
where
\begin{gather*}
\begin{aligned}
H_5&=[2A_0(A_1'+A_1P')+A_0B_2-2A_1(A_0'+A_0Q')]e^{P+Q}\\
&\quad +[2B_2(A_1'+A_1P')-A_1B_1]e^{2P}-[A_0''+2A_0'Q'+A_0(Q')^2\\
&\quad +A_0Q''-2(\frac{A_0'}{A_0}+Q')(A_0'+A_0Q')]e^Q\\
&\quad -[B_2(\frac{A_0''}{A_0}+2\frac{A_0'}{A_0}Q'+(Q')^2+Q'')-B_1(\frac{A_0'}{A_0}+Q')]e^P,
\end{aligned}
\\
H_4=A_1A_0e^{P+Q}+A_1B_2e^{2P}-2(A_0'+A_0Q')e^Q-B_1e^P,\\
B_1=A_1''+2A_1'P'+A_1(P')^2+A_1P''-\frac{A_1}{A_0}(A_0''+2A_0'Q'+A_0(Q')^2
+A_0Q''),
\\
B_2=A_1'+A_1P'-A_1(\frac{A_0'}{A_0}+Q').
\end{gather*}
Clearly, $B_1,B_2$ are meromorphic functions with $\sigma(B_j)<n$
($j=1,2$). By \eqref{e3.22}-\eqref{e3.24}, we see that
\[
-(\frac{\varphi''}{\varphi}+H_3\frac{\varphi'}{\varphi}+H_2)
=-\frac{1}{\varphi_2(z)}\{\frac{\varphi''}{\varphi}\varphi_2(z)
+\frac{\varphi'}{\varphi}H_4+H_5+A_0^2e^{2Q}\}.
\]
 As $\varphi_2(z)$,
$H_4$, $H_5$ have the form of $\psi_2(z)$ which is defined as in
lemma \ref{lem4} (ii), so by lemma \ref{lem4} (i) and (vi), we get
$\frac{\varphi''}{\varphi}\varphi_2(z)+\frac{\varphi'}{\varphi}H_4+H_5+A_0^2e^{2Q}\not\equiv
0$, $\varphi_2(z)\not\equiv 0$. Hence
$-(\varphi''+H_3\varphi'+H_2\varphi)\not\equiv0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
 First, we suppose $d_2\not\equiv 0$.
Suppose that $f \not\equiv 0$ is a solution of equation
 \eqref{e1.1}, by Theorem \ref{thmA} we have $\sigma(f)=\infty$.
Set $w=d_2f''+d_1f'+d_0f-\varphi$, then
$\sigma(w)=\sigma(g)=\sigma(f)=\infty$ by lemma \ref{lem3}.

To prove that $\overline{\lambda}(g-\varphi)=\infty$, we need to prove
only that $\overline{\lambda}(w)=\infty$.
Substituting $f''=-A_1e^Pf'-A_0e^Qf$ into  $w$, we  get
\begin{equation}
w=(d_1-d_2A_1e^{P})f'+(d_0-d_2A_0e^Q)f-\varphi. \label{e3.25}
\end{equation}
Differentiating both sides of equation \eqref{e3.25}, and replacing $f''$
with $f''=-A_1e^Pf'-A_0e^Qf$, we obtain
\begin{equation}
\begin{aligned}
w'&=[d_2A_1^2e^{2P}-((d_2A_1)'+P'd_2A_1+d_1A_1)e^P-d_2A_0e^Q+d_0+d_1']f'\\
&\quad +[d_2A_0A_1e^{P+Q}-((d_2A_0)'+Q'd_2A_0+d_1A_0)e^Q+d'_0]f-\varphi'.
\end{aligned}  \label{e3.26}
\end{equation}
Set
\begin{gather*}
\alpha_1=d_1-d_2A_1e^{P},\quad \alpha_0=d_0-d_2A_0e^Q,\\
\beta_1=d_2A_1^2e^{2P}-((d_2A_1)'+P'd_2A_1+d_1A_1)e^P-d_2A_0e^Q+d_0+d_1',\\
\beta_0=d_2A_0A_1e^{P+Q}-((d_2A_0)'+Q'd_2A_0+d_1A_0)e^Q+d'_0.
\end{gather*}
Then we have
\begin{gather*}
\alpha_1f'+\alpha_0f=w+\varphi\\
 \beta_1f'+\beta_0f=w'+\varphi'.
\end{gather*}
Set
\begin{equation}
\begin{aligned}
h&=\alpha_1\beta_0-\alpha_0\beta_1 \\
&=[d_1-d_2A_1e^{P}][d_2A_0A_1e^{P+Q}-((d_2A_0)'+Q'd_2A_0+d_1A_0)e^Q+d'_0]\\
&\quad -[d_0-d_2A_0e^Q][d_2A_1^2e^{2P}-((d_2A_1)'+P'd_2A_1+d_1A_1)e^P\\
&\quad -d_2A_0e^Q + d_0+d_1'].
\end{aligned}  \label{e3.27}
\end{equation}
 Now check all terms of $h$. Since the term $\pm d_2^2A_1^2A_0e^{2P+Q}$ is
 eliminated, by  \eqref{e3.27} we can write
\begin{equation}
h=\psi_2(z)-d_2^2A_0^2e^{2Q}, \label{e3.28}
\end{equation}
where $\psi_2(z)$ is defined as in lemma \ref{lem4} (ii). By $d_2\not\equiv 0,\ A_0\not\equiv 0$
and lemma \ref{lem4} (ii), we see
that $h\not\equiv 0$. By \eqref{e3.25} and \eqref{e3.26}, we obtain
\begin{equation}
\begin{aligned}
f'&=\frac{1}{h}\{-(d_0-d_2A_0e^Q)(w'+\varphi')\\
&\quad +[d_2A_0A_1e^{P+Q}-((d_2A_0)'+Q'd_2A_0+d_1A_0)e^Q+d'_0](w+\varphi)\}\\
&=\frac{1}{h}\{-(d_0-d_2A_0e^Q)w'+\Phi_{10}w+\varphi
d_2A_0A_1e^{P+Q}\\
&\quad +[d_2A_0\varphi'-((d_2A_0)'+Q'd_2A_0+d_1A_0)\varphi]e^Q+\psi_1\},
\end{aligned} \label{e3.29}
\end{equation}
  where $\Phi_{10}$ is an entire function with
$\sigma(\Phi_{10})\leq n$, $\psi_1$ is defined as in lemma \ref{lem4} (i).
\begin{equation}
\begin{aligned}
f&=\frac{1}{h}\{(d_1-d_2A_1e^{P})(w'+\varphi')\\
&\quad -[d_2A_1^2e^{2P}-((d_2A_1)'+P'd_2A_1+d_1A_1)e^P-d_2A_0e^Q+d_0+d_1'](w+\varphi)\}\\
&=\frac{1}{h}\{(d_1-d_2A_1e^{P})w'+\Phi_{00}w-\varphi
d_2A_1^2e^{2P}+\varphi d_2A_0e^Q+\psi_1\},
\end{aligned}  \label{e3.30}
\end{equation}
where $\Phi_{00}$ is an entire function with $\sigma(\Phi_{00})\leq
n$, $\psi_1$ is defined as in lemma \ref{lem4} (i).
Differentiating both sides of equation \eqref{e3.29}, and by \eqref{e3.28},
we get
\begin{equation}
f''=\frac{1}{h^2}\{(-d_2^3A_0^3e^{3Q}+\psi_3)w''+\Phi_{21}w'
 +\Phi_{20}w+\psi_4\}, \label{e3.31}
\end{equation}
where $\Phi_{21}$ and $\Phi_{20}$ are  entire functions with
$\sigma(\Phi_{21})\leq n$, $\sigma(\Phi_{20})\leq n$, $\psi_3$,
$\psi_4$ are defined as in lemma \ref{lem4} (iii)-(iv).
Substituting \eqref{e3.28}-\eqref{e3.31} into \eqref{e1.1}, we obtain
\begin{align*}
&(-d_2^3A_0^3e^{3Q}+\psi_3)w''+\Phi_{21}w'+\Phi_{20}w+\psi_4\\
&+A_1e^{P(z)}(\psi_2(z)-d_2^2A_0^2e^{2Q})\{-(d_0-d_2A_0e^Q)w'+\Phi_{10}w+\varphi
d_2A_0A_1e^{P+Q}\\
&+[d_2A_0\varphi'-((d_2A_0)'+Q'd_2A_0+d_1A_0)\varphi]e^Q+\psi_1\}\\
&+A_0e^{Q(z)}(\psi_2(z)-d_2^2A_0^2e^{2Q})\{(d_1-d_2A_1e^{P})w'\\
&+\Phi_{00}w-\varphi d_2A_1^2e^{2P}+\varphi d_2A_0e^Q+\psi_1\}=0,
\end{align*}
namely
\begin{equation}
(-d_2^3A_0^3e^{3Q}+\psi_3)w''+\Phi_1w'+\Phi_0w=F, \label{e3.32}
\end{equation}
where $\Phi_1$ and $\Phi_0$ are  entire functions with
$\sigma(\Phi_1)\leq n$, $\sigma(\Phi_0)\leq n$, and
\begin{equation}
\begin{aligned}
-F&=\psi_4+(A_1e^{P}\psi_2-d_2^2A_1A_0^2e^{(P+2Q)})(\varphi
d_2A_0A_1e^{P+Q}\\
&\quad +[d_2A_0\varphi'-((d_2A_0)'+Q'd_2A_0+d_1A_0)\varphi]e^Q+\psi_1))\\
&\quad +(A_0e^{Q}\psi_2-d_2^2A_0^3e^{3Q})(-\varphi
d_2A_1^2e^{2P}+\varphi d_2A_0e^Q+\psi_1)\\
&=\psi_4+A_1^2A_0\psi_2\varphi
d_2e^{2P+Q}+A_1\psi_2[d_2A_0\varphi'-(d_2A_0)'\varphi-Q'd_2A_0\varphi\\
&\quad -d_1A_0\varphi]e^{P+Q}+A_1e^{P}\psi_1\psi_2-d_2^2A_1A_0^2e^{(P+2Q)}
 \psi_1-d_2^2A_1A_0^2[d_2A_0\varphi'\\
&\quad -(d_2A_0)'\varphi-Q'd_2A_0\varphi-d_1A_0\varphi]e^{P+3Q}
  -d_2^3A_0^3A_1^2\varphi e^{2P+3Q} \\
&\quad -\varphi\psi_2 d_2 A_0A_1^2e^{2P+Q}+\varphi
d_2^3A_1^2A_0^3e^{2P+3Q}+\psi_2\varphi d_2A_0^2e^{2Q}\\
&\quad -\varphi d_2^3A_0^4e^{4Q}
  +A_0e^{Q}\psi_1\psi_2-d_2^2A_0^3e^{3Q}\psi_1.
\end{aligned} \label{e3.33}
\end{equation}
Since every $\psi_2$ in \eqref{e3.33} is equal to that in \eqref{e3.28}, so the
terms $\pm A_1^2A_0\psi_2\varphi d_2e^{2P+Q}$ and
$\pm \varphi d_2^3A_1^2A_0^3e^{2P+3Q}$ are eliminated. By lemma \ref{lem4} (iv),
we know that
\begin{gather*}
A_1\psi_2[d_2A_0\varphi'-(d_2A_0)'\varphi-Q'd_2A_0\varphi-d_1A_0\varphi]e^{P+Q},
\\
A_1e^{P}\psi_1\psi_2,\quad -d_2^2A_1A_0^2e^{(P+2Q)}\psi_1,\\
-d_2^2A_1A_0^2[d_2A_0\varphi'-(d_2A_0)'\varphi-Q'd_2A_0\varphi-d_1A_0\varphi]
e^{P+3Q},\\
\psi_2\varphi d_2A_0^2e^{2Q},\quad  A_0e^{Q}\psi_1\psi_2,\quad
-d_2^2A_0^3e^{3Q}\psi_1
\end{gather*}
 having all forms of $\psi_4$,  by \eqref{e3.33}, we obtain
\begin{equation}
-F=-\varphi d_2^3A_0^4e^{4Q}+\psi_4. \label{e3.34}
\end{equation}
By lemma \ref{lem4} (iii)-(iv) and
$d_2\not\equiv 0$,\ $\varphi\not\equiv 0$, $A_0\not\equiv 0$ and
$\sigma(\varphi)<n$, we see that
\begin{equation}
F\not\equiv 0,\quad  -d_2^3A_0^3e^{3Q}+\psi_3\not\equiv 0. \label{e3.35}
\end{equation}
By equation \eqref{e3.32}, lemma \ref{lem5}, $\sigma(w)=\infty$ and
\eqref{e3.35}, we
obtain $\overline{\lambda}(w)=\sigma(w)=\infty$.

Now suppose $d_2\equiv 0$, $d_1\not\equiv 0$, $d_0\not\equiv 0$.
Using a similar reasoning to that above, we get
$\overline{\lambda}(w)=\sigma(w)=\infty.$
Finally, if $d_2\equiv 0$, $d_1\not\equiv 0$, $d_0\equiv 0$ or
$d_2\equiv 0$,\ $d_1\equiv 0$, $d_0\not\equiv 0$, then for
$w=d_jf^{(j)}-\varphi\ (j=1$ or $0)$,  we can consider
$\frac{w}{d_j}=f^{(j)}-\frac{\varphi}{d_j}$. Since
$\overline{\lambda}(w)=\overline{\lambda}(\frac{w}{d_j})$
($d_j$ being polynomials), using a similar reasoning
as in  Theorem \ref{thm1} and $\sigma(w)=\infty$, we get
 $\overline{\lambda}(w)=\sigma(w)=\infty$.
\end{proof}


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\end{document}