\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 73, pp. 1--9.\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-2011/73\hfil Entire solutions]
{Entire solutions for a nonlinear differential equation}

\author[J. Qi, J. Ding, T. Zhu  \hfil EJDE-2011/73\hfilneg]
{Jianming Qi, Jie Ding, Taiying Zhu}  

\address{Jianming Qi \newline
Department of Mathematics and Physics, Shanghai Dianji University,
Shanghai 200240, China}
\email{qijianming1981@gmail.com}

\address{Jie Ding \newline
Department of Mathematics, Shandong University, 
Jinan 250100, China}
\email{dingjie169@163.com}

\address{Taiying Zhu \newline
Department of Mathematics and Physics, Shanghai Dianji University,
Shanghai 200240, China}
\email{ztyyyy@163.com}


\thanks{Submitted July 10, 2010. Published June 15, 2011.}
\thanks{Supported by project 10XKJ01 from Leading Academic Discipline 
Project of Shanghai Dianji \hfill\break\indent 
 University, and grants: 10771121 from the NSFC,
Z2008A01 from the NSF of Shandong, \hfill\break\indent
and 20060422049 from the RFDP}

\subjclass[2000]{30D35, 30D45}
\keywords{Transcendental entire functions;
 Nevanlinna theory; \hfill\break\indent differential equations}

\begin{abstract}
 In this article, we study the existence of solutions to
 the differential equation
 $$
 f^n(z)+P(f)= P_1e^{h_1}+ P_2e^{h_2},
 $$
 where $n\geq 2$ is an positive integer,  $f$ is a transcendental
 entire function, $P(f)$ is a differential polynomial in $f$ of
 degree less than or equal $n-1$,  $P_1, P_2$ are  small functions 
 of $e^z$,  $h_1$, $h_2$ are polynomials, and $z$ is in the open 
 complex plane $\mathbb{C}$.
 Our results extend those obtained by  Li \cite{L1,L2,L3,L4}.
\end{abstract}

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

\section{Introduction and main results}

 Nevanlinna value distribution theory of meromorphic functions has been
extensively applied to resolve growth (see\cite{L1}), value
distribution \cite{L1}, and solvability of meromorphic
solutions of linear and nonlinear differential equations
\cite{H2,L1,Y1,Y2}. Considering
meromorphic functions $f$ in the complex plane, we assume that the
reader is familiar with the standard notations and results such as
the proximity function $m(r, f)$, counting function $N(r, f)$,
characteristic function $T(r, f)$, the first and second main
theorems, lemma on the logarithmic derivatives etc. of Nevanlinna
theory; see e.g. \cite{H1,L1}. Given a meromorphic function
$f$, we shall call a meromorphic function $a(z)$ a small function of
$f(z)$ if $T(r,a)=S(r,f)$, where $S(r, f)$ is used to denote any
quantity that satisfies $S(r, f)=o(T(r, f))$ as
$r\to\infty$, possibly outside  a set of $r$ of finite
logarithmic measure.  A differential polynomial $P(f)$ in $f$ is a
polynomial in $f$ and its derivatives with small functions of $f$ as
the coefficients. The notation $\mathscr{F}$ is defined to the
family of all meromorphic functions which satisfy
$\overline{N}(r,\frac{1}{h})+\overline{N}(r, h)=S(r, h)$. Note that
all functions in family $\mathscr{F}$ are transcendental, and all
functions of the form $be^{\lambda z}$ are functions in family
$\mathscr{F}$, where $\lambda$ is any nonzero constant and $b$ is a
rational function.

In 2006,  Li and  Yang \cite{L2,Y2} obtain the
following results.

\begin{theorem} \label{thmA}
Let $n\geq 4$ be an  integer, and $P(f)$
denote an algebraic differential polynomial in $f$ of degree
$\leq n-3$. Let $P_1$, $P_2$ be two nonzero polynomials,
$\alpha_1$ and $\alpha_2$ be two nonzero constants with
$\frac{\alpha_1}{\alpha_2}\neq $ rational. Then the differential
equation
$$
f^n(z)+P(f)= P_1e^{\alpha_1z}+ P_2e^{\alpha_2z}
$$
 has no transcendental entire solutions.
\end{theorem}

\begin{theorem} \label{thmB}
 Let $n\geq 3$ be an  integer, and $P(f)$ be
an algebraic differential polynomial in $f$ of degree $\leq n-3$,
$b(z)$ be a meromorphic function, and $\lambda$, $c_1$, $c_2$ and
three nonzero constants, Then the differential equation
$$
f^n(z)+P(f)= b(z)(c_1e^{\lambda z}+c_2e^{-\lambda z})
$$
 has no transcendental entire solutions $f(z)$,
satisfying $T(r, b)= S(r, f)$.
\end{theorem}

Recently, Considering  the  degree of the differential polynomial
$P(f)$ of $n-2$ or $ n-1$, P. Li \cite{L4} proved  the following
results which
are improvements or complementarity of Theorems \ref{thmA} and
\ref{thmB}.

\begin{theorem} \label{thmC}
Let $n\geq 2$ be an integer. Let $f$ be a
transcendental entire function, $P(f)$ be a differential polynomial
in $f$ of degree $\leq n-1$. If
\begin{equation}
f^n(z)+P(f)= P_1e^{\alpha_1z}+ P_2e^{\alpha_2z},\label{e1.1}
\end{equation}
where $P_i(i=1, 2)$ are nonvanshing small functions of $e^z$,
$\alpha_i (i=1, 2)$ are positive numbers satisfying
$(n-1)\alpha_2\geq n\alpha_1>0$, then there exists a small function
$\gamma$ of $f$ such that
\begin{equation}
(f-\gamma)^n= P_2e^{\alpha_2z}.\label{e1.2}
\end{equation}
\end{theorem}



\begin{theorem} \label{thmD}
 Let $n\geq 2$ be an  integer, $\alpha_1, \alpha_2$
be real numbers and $\alpha_1<0<\alpha_2$. Let $P_1$,
$P_2$ be small functions of $e^z$. If there exists a transcendental
entire function $f$ satisfying the differential equation \eqref{e1.1},
where $P(f)$ is a differential polynomial in $f$ of degree not
exceeding $n-2$, then $\alpha_1+\alpha_2=0$, and there exist
constants $c_1, c_2$ and small functions $\beta_1, \beta_2$ with
respect to $f$ such that
\begin{equation}
f=c_1\beta_1e^{\alpha_1z/n} +c_2\beta_2e^{\alpha_2z/n},\label{e1.3}
\end{equation}
moreover, $\beta_i^n=P_i$, $i=1, 2$.
\end{theorem}


\begin{theorem} \label{thmE}
 Let $n\geq 2$ be an  integer, $\alpha_1$, $\alpha_2$
 be positive numbers satisfying $(n-1)\alpha_2\geq
n\alpha_1>0$. Let $P_1$, $P_2$ be small functions of $e^z$. If
$\frac{\alpha_1}{\alpha_2}$ is irrational, then the differential
equation \eqref{e1.1} has no entire solutions, where $P(f)$ is
a differential polynomial in $f$ of degree $\leq n-1$.
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
By an  example, Li  \cite{L4} pointed  if the degree of $P(f)$
is $n-1$, then the solutions of \eqref{e1.1} may not be the
form in \eqref{e1.3}.
\end{remark}

It is  natural to ask whether $\alpha_1z$ and $\alpha_2z$
in \eqref{e1.1} can be replaced by two polynomials.
In this article, by the same method as in \cite{L4},
 we obtain the following results.

\begin{theorem} \label{thm1}
 Let $n\geq 2$ be an  integer. Let $f$ be a
transcendental entire function, $P(f)$ be a differential polynomial
in $f$ of degree $\leq n-1$. If
\begin{equation}
f^n(z)+P(f)= P_1e^{Q_1(z)}+ P_2e^{Q_2(z)},\label{e1.4}
\end{equation}
where $P_i(i=1, 2)$ are nonvanshing small meromorphic functions of
$e^z$, $Q_1(z)=\alpha_kz^k+
\alpha_{k-1}z^{k-1}+\dots+\alpha_1z+\alpha_0$,
$Q_2(z)=\beta_kz^k+ \beta_{k-1}z^{k-1}+\dots+\beta_1z+\beta_0$
are two polynomials satisfying $(n-1)\beta_k\geq n\alpha_k>0$
(where $\alpha_{k-1},\dots \alpha_0, \beta_{k-1},\dots \beta_0$ are finite constants and $k\geq
1$ ) is a positive integer, then there exists a small meromorphic
function $\gamma$ of $f$ such that
\begin{equation}
(f-\gamma)^n= P_2e^{Q_2}.\label{e1.5}
\end{equation}
\end{theorem}

\begin{theorem} \label{thm2}
 Let $n\geq 2$ be an integer and $P_1$, $P_2$
be small functions of $e^z$. If there exists a transcendental entire
function $f$ satisfying the differential equation \eqref{e1.4}, where
$P(f)$ is a differential polynomial in $f$ of degree not exceeding
$n-2$ and $\alpha_k<0<\beta_k$, then $\alpha_k+\beta_k=0$, and there
exist constants $c_1, c_2$ and small functions $\beta_1, \beta_2$
with respect to $f$ such that
$$
f=c_1\beta_1e^{\frac{Q_1}{n}}+c_2\beta_2e^{\frac{Q_2}{n}},\label{e1.6}
$$
moreover, $\beta_i^n=P_i$, $i=1, 2$.
\end{theorem}

\begin{theorem} \label{thm3}
 Let $n\geq 2$ be an  integer,
  $P_1$, $P_2$ be small functions of $e^z$. If
$\frac{\alpha_k}{\beta_k}$ is irrational, then the differential
equation \eqref{e1.4} has no entire solutions, where $P(f)$ is a
differential polynomial in $f$ of degree $\leq n-1$ and
$(n-1)\beta_k\geq n\alpha_k>0$.
\end{theorem}

Obviously, our results generalize the results in \cite{L1,L2,L3,L4}.

\section{Preliminary Lemmas}

In order to prove our theorems, we need the following lemmas.
First, we need the following well-known Clunie's lemma, which has
been extensively applied in studying the value distribution of a
differential polynomial $P(z, f)$, as well as the growth estimates
of solutions and meromorphic solvability of differential equations
in the complex plane.


\begin{lemma}[\cite{B,C}] \label{lem2.1}
Let $f$ be a transcendental meromorphic solution of
$$
f^nA(z, f)=B(z, f),
$$
where $A(z,f)$, $B(z, f)$ are differential polynomials in $f$ and
its derivatives with small meromorphic coefficients $a_{\lambda}$,
in the sense of $T(r, a_{\lambda})=S(r, f)$ for all $\lambda\in I$,
where $I$ is an index set. If the total degree of $B(z,f)$ as a
polynomial in $f$ and its derivatives is less than or equal  $n$,
then $m(r, A(z,f))=S(r,f)$.
\end{lemma}


\begin{lemma}[\cite{H1}] \label{lem2.2}
Suppose that $f$ is a nonconstant meromorphic function and
$F= f^n + Q(f)$, where $Q(f)$ is a
differential polynomial in $f$ with degree $\leq n-1$. If
$N(r,f)+N(r,\frac{1}{F})=S(r,f)$, then
$$
F=(f+\gamma)^n,
$$
whereby $\gamma$ is meromorphic and $T(r,\gamma)=S(r, f)$
\end{lemma}

\begin{lemma}[\cite{L3}] \label{lem2.3}
Suppose that $h$ is a function in family $\mathscr{F}$.
Let $f=a_0h^p+a_1h^{p-1}+\dots+a_p$, and
$g=b_0h^q+b_1h^{q-1}+\dots+b_q$ be polynomials in $h$ with all
coefficients being small functions of $h$ and $a_0b_0a_p\neq 0$
If $q\leq p$, then $m(r,\frac{g}{f})=S(r,h)$.
\end{lemma}

\section{Proofs of main theorems}

\begin{proof}[Proof of Theorem \ref{thm1}]
 First of all, we write $P(f)$ as
\begin{equation}
P(f)=\sum_{j=0}^{n-1}b_jM_j(f),\label{e3.1}
\end{equation}
where $b_j$ are small functions of $f$, $M_0(f)=1$,
$M_j(f) (j=1, 2,\dots , n-1)$ are homogeneous differential
monomials in $f$ of degree $j$. Without loss of generality,
we assume that $b_0\not\equiv 0$,
otherwise, we do the transformation $f=f_1+c$ for a suitable
constant $c$. From \eqref{e1.4}, we have
\begin{equation}
\frac{1}{P_1e^{Q_1}+P_2e^{Q_2}-b_0}
+\sum_{j=1}^{n-1}\frac{b_j}{P_1e^{Q_1}+P_2e^{Q_2}-b_0}
\frac{M_j(f)}{f^j}(\frac{1}{f})^{n-j}=(\frac{1}{f})^n.\label{e3.2}
\end{equation}
Note that $m(r, \frac{M_j(f)}{f^j})=S(r,f)$,
\begin{align*}
&m(r, \frac{1}{P_1e^{Q_1(z)}+P_2e^{Q_2(z)}-b_0})\\
&= m(r,\frac{1}{P_1e^{\alpha_{k-1}z^{k-1}+\dots+\alpha_0}
e^{\alpha_kz^k}+P_2e^{\beta_{k-1}z^{k-1}+\dots
+\beta_0}e^{\beta_kz^k}-b_0}),
\end{align*}
where $P_1$, $P_2$, $e^{\alpha_{k-1}z^{k-1}+\dots+\alpha_0}$,
$e^{\beta_{k-1}z^{k-1}+\dots+\beta_0}$ are small functions of
$e^{z^k}$.

We take $h=e^{z^k}$, $q=0, p=\beta_k$, by Lemma \ref{lem2.3}, we obtain
\begin{align*}
&m(r, \frac{1}{P_1e^{Q_1(z)}+P_2e^{Q_2(z)}-b_0})\\
&=S(r, e^{z^k})=S(r,P_1e^{Q_1(z)}+P_2e^{Q_2(z)}-b_0)
=S(r, f(z)).
\end{align*}
Therefore, the left-hand side of \eqref{e3.2} is a polynomial in
$1/f$ of degree at most $n-1$ with coefficients being small
proximate functions of $1/f$. Hence
\begin{equation}
m(r,\frac{1}{f})=S(r, f).\label{e3.3}
\end{equation}
Taking the derivatives in both sides of \eqref{e1.4} gives
\begin{equation}
nf^{n-1}f'+(P(f))'=(P_1'+Q_1'P_1)e^{Q_1}+(P'_2+Q_2'P_2)e^{Q_2}.
\label{e3.4}
\end{equation}
By eliminating $e^{Q_1}$ and  $e^{Q_2}$, respectively
from \eqref{e1.4} and the above equation, we obtain
\begin{gather}
(P_2'+Q_2'P_2)f^n-P_2nf^{n-1}f'+(P_2'+Q_2'P_2)P(f)-P_2(P(f))'=\beta
e^{Q_1}\label{e3.5}
\\
(P_1'+Q_1'P_1)f^n-P_1nf^{n-1}f'+(P_1'+Q_1'P_1)P(f)-P_1(P(f))'=-\beta
e^{Q_2},\label{e3.6}
\end{gather}
where $\beta= P_1P_2'-P_2P_1'+(Q'_2-Q'_1)P_1P_2$ which is a small
function of $f$. We note that $\beta$ cannot vanish identically,
otherwise, by integration we obtain $e^{Q_2-Q_1}=C\frac{P_1}{P_2}$ for
a constant, which is impossible. From \eqref{e3.5} and \eqref{e3.6},
we obtain
\begin{equation}
m(r, e^{Q_j})\leq n T(r, f)+S(r,f),\quad  j=1,2.\label{e3.7}
\end{equation}
On the other hand, from \eqref{e1.4}, we have
\begin{equation}
nT(r,f)=m(r, f^n)=m(r, f^n+P(f))\leq T(r, P_1e^{Q_1}+P_2e^{Q_2})+
S(r,f).\label{e3.8}
\end{equation}
Therefore, $S(r, e^{Q_1})=S(r, e^{Q_2})=S(r, f):=S(r)$.
From \eqref{e3.2},
we have
$$
\frac{e^{Q_i}}{p_1e^{Q_1}+p_2e^{Q_2}-b_0}
+\sum_{j=1}^{n-1}\frac{b_je^{Q_i}}{p_1e^{Q_1}+p_2e^{Q_2}-b_0}
\frac{M_j(f)}{f^j}\frac{1}{f^{n-j}}=\frac{e^{Q_i}}{f^n},\quad i=1, 2.
$$
It follows that
\begin{equation}
m(r,\frac{e^{Q_i}}{f^n})=S(r),\quad  i=1, 2.\label{e3.9}
\end{equation}
Next, we prove
\begin{equation}
m(r, \frac{e^{Q_1}}{f^{n-1}})=S(r).\label{e3.10}
\end{equation}
For a fixed $r>0$, let $z=re^{i\theta}$. The interval $[0, 2\pi)$
can be expressed as the union of the following three disjoint sets:
\begin{gather*}
E_1=\{\theta\in[0, 2\pi)|\frac{|f(z)|}{|e^{Q_2(z)-Q_1(z)}|}\leq 1\},\\
E_2=\{\theta\in[0, 2\pi)|\frac{|f(z)|}{|e^{Q_2(z)-Q_1(z)}|}> 1,
|e^{z^k}|\leq 1\}, \\
E_3=\{\theta\in[0, 2\pi)|\frac{|f(z)|}{|e^{Q_2(z)-Q_1(z)}|}> 1,
|e^{z^k}|> 1\}.
\end{gather*}
By the definition of the proximate function, we have
$$
m(r,\frac{e^{Q_1(z)}}{f^{n-1}(z)})
=\frac{1}{2\pi}\int_{0}^{2\pi}\log^+|\frac{e^{Q_1(z)}}{f^{n-1}(z)}
|d\theta= I_1+ I_2 + I_3,
$$
where
$$
I_j=\frac{1}{2\pi}\int_{E_j}\log^+|\frac{e^{Q_1(z)}}{f^{n-1}(z)}|
d\theta,\quad (j=1, 2, 3).
$$
For $\theta\in E_1$, we have $|f(z)|\leq |e^{Q_2(z)-Q_1(z)}|$. Since
$\frac{e^{Q_1(z)}}{f^{n-1}(z)}
=\frac{e^{Q_2(z)}}{f^{n}(z)}\frac{f(z)}{e^{Q_2(z)-Q_1(z)}}$,
we obtain
$$
I_1\leq m(r,\frac{e^{Q_2}}{f^n})=S(r).
$$
For $\theta\in E_2$, we have
$|e^{Q_1(z)}|=|e^{\alpha_k z^k(1+o(1))}|\leq 1$, and thus
$|\frac{e^{Q_1(z)}}{f^{n-1}(z)}|\leq\frac{1}{|f^{n-1}(z)|}$. It
follows from \eqref{e3.3} that
$$
I_2\leq m(r,\frac{1}{f^{n-1}})= S(r).
$$
For $\theta\in E_3$, we have $|f(z)|>|e^{Q_2(z)-Q_1(z)}|$.
Therefore,
\begin{align*}
|\frac{e^{Q_1(z)}}{f^{n-1}(z)}|
&\leq\frac{|e^{Q_1(z)}|}{|e^{(n-1)(Q_2(z)-Q_1(z))}|}\\
& =\frac{1}{|e^{(n-1)Q_2(z)-nQ_1(z)}|}
=\frac{1}{|e^{((n-1)\beta_k-n\alpha_k)z^k(1+o(1))}|}.
\end{align*}
By the assumption $(n-1)\beta_k\geq n\alpha_k>0$, we obtain
$|\frac{e^{Q_1(z)}}{f^{n-1}(z)}|\leq 1$. Therefore, we have $I_3=0$.
Hence \eqref{e3.10} holds.

It follows from \eqref{e3.5} that
$$
f^{n-1}\varphi=\beta\frac{e^{Q_1}}{f^{n-1}}f^{n-1}-R(f),
$$
where $\varphi=(P_2'+P_2Q_2')f-nP_2f'$, and
$$
R(f)=(P_2'+P_2Q_2')P(f)-P_2P'(f)
$$
which is a differential polynomial in $f$ of degree at most $n-1$.
By Lemma \ref{lem2.1}, we obtain $m(r, \varphi)=S(r, f)$. Note that since
$\varphi$ is entire, we have $N(r, \varphi)=S(r, \varphi)=S(r, f)$.
Hence $T(r, \varphi)= S(r, f)$, i.e., $\varphi$ is a small function
of $f$, By the definition of $\varphi$, we obtain
$$
f'=\frac{P_2'+Q'_2P_2}{nP_2}f-\frac{\varphi}{nP_2}.
$$
Substituting the above equation into \eqref{e3.6} gives
$$
f^n-\frac{n
P_1\varphi}{\beta}f^{n-1}-\frac{P_2(P'_1+Q_1'P_1)}{\beta}P(f)
+\frac{P_1P_2}{\beta}(P(f))'=P_2e^{Q_2}.
$$
By Lemma \ref{lem2.2}, we see that there exists a small function $\gamma$ of
$f$ such that $(f-\gamma)^n=P_2e^{Q_2}$. This also completes the
proof of Theorem \ref{thm1}.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2}]
 We  discuss only the case $\alpha_k+\beta_k\geq 0$.
The case $\alpha_k+\beta_k\leq 0$ can be discussed similarly.
Suppose that $f$ is a transcendental entire solution of \eqref{e1.4}.
Similar to the proof of Theorem \ref{thm1}, we can still
get \eqref{e3.3}-\eqref{e3.9}. For
a fixed $r>0$, let $z=re^{i\theta}$. We can express the interval
$[0, 2\pi)$ as the union of the following three disjoint sets:
\begin{gather*}
E_1=\{\theta\in [0, 2\pi)|\frac{|f^2(z)|}{|e^{Q_2(z)-Q_1(z)}|}\leq
1|\}, \\
E_2=\{\theta\in[0, 2\pi)|\frac{|f^2(z)|}{|e^{Q_2(z)-Q_1(z)}|}> 1,
|e^{z^k}|\leq 1\},\\
E_3=\{\theta\in[0, 2\pi)|\frac{|f^2(z)|}{|e^{Q_2(z)-Q_1(z)}|}> 1,
|e^{z^k}|> 1\}.
\end{gather*}
By the definition of the proximate function, we have
$$
m(r,\frac{e^{Q_1(z)+Q_2(z)}}{f^{2n-2}(z)})=\frac{1}{2\pi}
\int_{0}^{2\pi}\log^+|\frac{e^{Q_1(z)+Q_2(z)}}{f^{2n-2}(z)}|d\theta
=I_1+ I_2+ I_3,
$$
where
$$
I_j=\frac{1}{2\pi}\int_{E_j}\log^+|\frac{e^{Q_1(z)
+Q_2(z)}}{f^{2n-2}(z)}|d\theta,\quad j=1, 2, 3.
$$
For $\theta\in E_1$, we have
$$
|\frac{e^{Q_1(z)+Q_2(z)}}{f^{2n-2}(z)}|
=|\frac{e^{2Q_2(z)}}{f^{2n}(z)}\frac{f^2(z)}{e^{Q_2(z)-Q_1(z)}}|
\leq |\frac{e^{Q_2(z)}}{f^n(z)}|^2.
$$
Thus by \eqref{e3.9}, we obtain $I_1\leq S(r)$. For $\theta\in E_2$,
it follows from $|e^{z^k}|\leq 1$ and $\alpha_k+\beta_k\geq 0$ that
$|e^{(\alpha_k+\beta_k)z^k(1+o(1))}|\leq 1$. Therefore,
$$
|\frac{e^{Q_1(z)+Q_2(z)}}{f^{2n-2}(z)}|\leq\frac{1}{|f^{2n-2}(z)|}.
$$
Then by \eqref{e3.3}, we obtain $I_2\leq S(r)$.
For $\theta\in E_3$, we have
$|f^2(z)|>|e^{Q_2(z)-Q_1(z)}|$. Thus
\begin{align*}
|\frac{e^{Q_1(z)+Q_2(z)}}{f^{2n-2}(z)}|
&<\frac{|e^{Q_1(z)+Q_2(z)}|}{|e^{(n-1)(Q_2(z)-Q_1(z))}|}
=\frac{1}{|e^{(n-2)Q_2(z)-nQ_1(z)}|}\\
&=\frac{1}{|e^{[(n-2)\beta_k -n\alpha_k]z^k(1+o(1))}|}\leq 1.
\end{align*}
It follows that $I_3\leq S(r)$. Hence we have
\begin{equation}
m(r,\frac{e^{Q_1+Q_2}}{f^{2n-2}})= S(r,f).\label{e4.1}
\end{equation}
Multiplying \eqref{e3.5} by \eqref{e3.6} gives
\begin{equation}
f^{2n-2}\varphi+Q(f)=-\beta^2e^{Q_1+Q_2},\label{e4.2}
\end{equation}
where $Q(f)$ is a differential polynomial in $f$ of degree at most
$2n-2$, and
\begin{equation}
\varphi=((P_1'+Q_1'P_1)f-nP_1f')((P_2'+Q_2'P_2)f-nP_2f')).\label{e4.3}
\end{equation}
From \eqref{e4.2} and by Lemma \ref{lem2.1}, we obtain $m(r,\varphi)=S(r, f)$.
Therefore, $T(r,\varphi)=S(r, f)$.

If $(P_1'+Q_1'P_1)f-nP_1f'\equiv 0$, then by integration we obtain
$f^n=cP_1e^{Q_1}$, for a nonzero constant $c$. Therefore,
$f=ae^{\frac{Q_1}{n}}$ for a small function $a$ of $f$. Thus we see
that the left-hand side of \eqref{e1.4} is a polynomial in
$e^{\frac{Q_1}{n}}$ of degree $n$. However, the right-hand side of
\eqref{e1.4} cannot be a polynomial in $e^{\frac{Q_1}{n}}$. Hence
$(P_1'+Q_1'P_1)f-nP_1f'\not\equiv 0$. Similarly, we have
$(P_2'+Q_2'P_2)f-nP_2f'\not\equiv 0$. Therefore, $\varphi\not\equiv
0$.
Let
\begin{equation}
(P_2'+Q_2'P_2)f-nP_2f'=h.\label{e4.4}
\end{equation}
Then we have
\begin{equation}
(P_1'+Q_1'P_1)f-nP_1f'=\frac{\varphi}{h}.\label{e4.5}
\end{equation}
By eliminating $f'$ and $f$, respectively from \eqref{e4.4}
and \eqref{e4.5}, we obtain
\begin{gather}
f=\frac{P_1}{\beta}h-\frac{\varphi P_2}{\beta}\frac{1}{h},\label{e4.6}
\\
f'=\frac{P_1'+Q_1'P_1}{n\beta}h-\frac{P_2'+Q_2'P_2}{n\beta}
\frac{\varphi}{h},\label{e4.7}
\end{gather}
where $\beta= P_1 P'_2-P_2 P'_1+(Q'_2-Q'_1)P_1 P_2$ which is a small
function of $f$, and cannot vanish identically.
From \eqref{e4.6}, we see that
$$
2T(r,h)=T(r,f)+S(r,f).
$$
Therefore, any small function of $f$ is also a small function of
$h$. And from the definition of $\varphi$ we see that $h$ is a
function in family $\mathscr{F}$. Thus $\frac{h'}{h}$ is a small
function of $f$. By taking derivative in both sides of \eqref{e4.6},
we obtain
\begin{equation}
f'=((\frac{P_1}{\beta})'+\frac{P_1}{\beta}\frac{h'}{h})h
-((\frac{\varphi P_2}{\beta})'-\frac{\varphi
P_2}{\beta}\frac{h'}{h})\frac{1}{h}.\label{e4.8}
\end{equation}
Comparing the coefficients of the right-hand side of \eqref{e4.7} and
\eqref{e4.8}, we deduce that
\begin{gather}
\frac{P_1'+Q'_1P_1}{n\beta}=(\frac{P_1}{\beta})'
+\frac{P_1}{\beta}\frac{h'}{h},\label{e4.9}\\
\frac{(P_2'+Q'_2P_2)\varphi}{n\beta}=(\frac{\varphi
P_2}{\beta})'-\frac{\varphi P_2}{\beta}\frac{h'}{h}.\label{e4.10}
\end{gather}
By integrating \eqref{e4.9} and \eqref{e4.10}, respectively,
we obtain
\begin{equation}
P_1e^{Q_1}=d_1(\frac{P_1}{\beta}h)^n,\quad
P_2e^{Q_2}=d_2(\frac{\varphi P_2}{\beta}\frac{1}{h})^n,\label{e4.11}
\end{equation}
where $d_1$ and $d_2$ are two nonzero constants. From the above two
equations, we see that there exist two small functions $\beta_1$ and
$\beta_2$ of $e^z$ such that $P_i=\beta_i^n$, $i=1, 2$, and
\begin{equation}
P_1P_2e^{Q_1+Q_2}=d_1d_2(\frac{P_1P_2\varphi}{\beta^2})^n.\label{e4.12}
\end{equation}
The right-hand side of the above equation is a small function of
$f$, and thus a small function of $e^{z^k}$. Therefore, the above
equation holds only when $\alpha_k+\beta_k\equiv 0$. Furthermore,
from \eqref{e4.11}, we see that there exist two nonzero
constants $c_1$ and $c_2$ such that
\begin{equation}
\frac{P_1}{\beta}h=c_1\beta_1e^{\frac{Q_1}{n}},\quad
\frac{P_2\varphi}{\beta}\frac{1}{h}
=-c_2\beta_2e^{\frac{Q_2}{n}}.\label{e4.13}
\end{equation}
Finally, from \eqref{e4.6}, we obtain \eqref{e1.6}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
If $f$ is a transcendental entire solution of \eqref{e1.4}, then by
 Theorem \ref{thm1}, there exists a small function $\gamma$ of $f$ such that
 \eqref{e1.5} holds. And thus $N(r, \frac{1}{f-\gamma})= S(r, f)$, i.e.,
 $\gamma$ is an exceptional small function of $f$. Equation
\eqref{e1.5} also  shows that there exist two small functions
$\omega_1$ and
 $\omega_2$ of $f$ such that $f'=\omega_1f+\omega_2$. By
 substituting this equation into \eqref{e1.4}, we see that
$P_1e^{Q_1}$ is  a polynomial in $f$ of degree $t< n$.
By Lemma \ref{lem2.2}, there exist two
 small functions $a$ and $\gamma_1$ of $f$ such that
 $$
a(f-\gamma_1)^t=P_1e^{Q_1}.\label{e5.1}
 $$
Therefore, $\gamma_1$ is also an exceptional small function of $f$.
Since any transcendental entire function cannot have two exceptional
small functions, we deduce that $\gamma_1=\gamma$.
From \eqref{e1.5} and above equation, we obtain
\begin{equation}
e^{nQ_1-tQ_2}=\frac{P_2^ta^n}{P_1^n}.\label{e5.2}
\end{equation}
The right-hand side of the above equation is small function of $f$,
and thus a small function of $e^z$. Hence we obtain
$nQ_1-tQ_2\equiv 0$. Therefore,
$\lim_{z\to\infty}\frac{Q_1}{Q_2}=\frac{\alpha_k}{\beta_k}
=\frac{t}{n}$
must be a rational number, which contradicts the assumption. This
also completes the proof of Theorem \ref{thm3}.
\end{proof}

\subsection*{Acknowledgements}
 The authors would like to express their
hearty thanks to Professor Hongxun Yi for his valuable advice and
helpful information.

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\end{document}