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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 94, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/94\hfil Growth of entire solutions]
{Growth of entire solutions of algebraic differential equations}

\author[W. Yuan, Y. Li, J. Lin \hfil EJDE-2012/94\hfilneg]
{Wenjun Yuan, Yezhou Li, Jianming Lin}  

\address{Wenjun Yuan \newline
School of Mathematics and information Science, Guangzhou University
Guangzhou 510006,  China \newline
Key Laboratory of Mathematics and Interdisciplinary Sciences,
Guangdong Higher Education Institutes\\
Guangzhou 510006, China}
\email{wjyuan1957@126.com}

\address{Yezhou Li \newline
School of science, 
Beijing University of Posts and Telecommunications\\
Beijing 100876, China}
\email{yiyexiaoquan@yahoo.com.cn}

\address{Jianming Lin \newline
School of Economic and Management, 
Guangzhou University of Chinese Medicine\newline
Guangzhou 510006,  China}
\email{ljmguanli@21cn.com}

\thanks{Submitted April 14, 2012. Published June 10, 2012.}
\thanks{Supported by grants 11101048 and 10771220 from the
NSF of China, 200810780002 from \hfill\break\indent
 the Doctorial Point Fund of National
Education Ministry of China, and 020586 from the 
\hfill\break\indent NSF of Guangdong}
\subjclass[2000]{34A20, 30D35}
\keywords{Entire function; normal family; algebraic
differential equations}

\begin{abstract}
 In this article, by means of the normal family theory, we estimate
 the growth order of entire solutions of some algebraic differential 
 equations and extend the  result by Qi et al \cite{QLY}.
 We also give some examples to show that our results occur in some 
 special cases.
\end{abstract}

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


\section{Introduction and main results} 

 Let $f(z)$ be a  holomorphic function on the complex plane. We use
the standard notation of Nevanlinna theory and denote the order of
$f(z)$ by $\rho(f)$ (see Hayman \cite{s7}, He \cite{s8}, Laine \cite{s9} and Yang \cite{s10}).
Let $D$ be a domain in the complex plane. A family $\mathcal{F}$ of
meromorphic functions in $D$ is normal, if each sequence
$\{f_n\}\subset \mathcal{F}$ contains a subsequence which converges
locally uniformly by spherical distance to a function $g(z)$
meromorphic in $D$ ($g(z)$ is permitted to be identically infinity).

We define spherical derivative of the  meromorphic function $f(z)$
as follows
$$
f^{\sharp}(z):=\frac{|f'(z)|}{1+|f(z)|^2}.
$$
An algebraic differential equation for $w(z)$ is of the form
\begin{equation}
P(z,w,w',\dots,w^{(k)})=0, \label{e1.1}
\end{equation}
where $P$ is a polynomial in each of its variables.

It is one of the important and interesting subjects to study the
growth of meromorphic the solution $w(z)$ of differential equation \eqref{e1.1}
in the complex plane.

In 1956, Goldberg \cite{s6} proved that the meromorphic solutions have
finite growth order when $k=1$. Some alternative proofs of this
result have been given by Bank and Kaufman \cite{s1}, by
Barsegian \cite{s2}.

In 1998, Barsegian \cite{B1,s3}  introduced an essentially new type of weight for
differential monomial below and gave the estimates first time for the growth order of meromorphic solutions of large classes of complex differential
equations of higher degrees by using his initial method  \cite{B2}. Later Bergweiler \cite{s4} reproved Barsegian's result
by using Zalcman's Lemma.

To state the result, we first introduce some notation \cite{B1}:
$n\in \mathbb{N}=\{1,2,3,\dots \}$,
$t_j\in \mathbb{N}_0=\mathbb{N}\bigcup \{0\}$ for $j=1,2,\dots,n$, and put
$t=(t_1,t_2,\dots,t_n)$. Define $M_t[w]$ by
$$
M_t[w](z):=[w'(z)]^{t_1}[w''(z)]^{t_2}\dots [w^{(n)}]^{t_n},
$$
with the convention that $M_{\{0\}}[w]=1$. We call
$p(t):=t_1+2t_2+\dots +nt_n$ the weight of $M_t[w]$. A differential
polynomial $P[w]$ is an expression of the form
\begin{equation}
P[w](z):=\sum_{t\in I}a_t(z,w(z))M_t[w](z), \label{e1.2}
\end{equation}
where the $a_{t}$ are rational in two variables and $I$ is a finite
index set. The total weight $W(P)$ of $P[w]$ is given by
$W(P):=\max_{t\in I}p(t)$.

\begin{definition} \label{def1} \rm
$\deg_{z,\infty} a_t$ denotes the degree
at infinity in variable $z$ concerning $a_t(z,w)$.
$\deg_{z,\infty} a:=\max_{t\in I}\max\{\deg_{z,\infty} a_t,0\}$.
\end{definition}

In 2009, the general estimates of growth order of meromorphic solutions $w(z)$
of the same equations, which depend on the degrees at infinity of
coefficients of differential polynomial in $z$, by Yuan et al
\cite{s11,s12}.

\begin{theorem}[\cite{s11}] \label{thmA}
 Let $w(z)$ be meromorphic  in the
complex plane and let $P[w]$ be a differential polynomial. If $w(z)$
satisfies the differential equation $ [w'(z)]^n=P[w] $ where  $n\in
\mathbb{N}$ and $n > W(P)$, then
the growth order $\rho:=\rho(w)$ of $w(z)$ satisfies
$$
\rho \leq 2+\frac{2\deg_{z,\infty} a}{n-W(P)}.
$$.
\end{theorem}

Barsegian \cite{s3} and Bergweiler \cite{s4} proved
$\rho < \infty$ under the same conditions as Theorem \ref{thmA}.
 Gu et al \cite{GLY} considered the case of entire solutions.
In 2012, Qi et al \cite{QLY} gave a small upper bound of the
growth order of the entire solutions.

\begin{theorem} \label{thmB}
Let $k, q, n\in \mathbb{N}$, and $P[w]$
be a differential polynomial with the form \eqref{e1.2}. Suppose that
$w(z)$ is an entire function whose all zeros  have multiplicity at
least and $nkq>W(P)$. If $w(z)$ satisfies the differential
equation $[Q(w^{(k)}(z))]^n=P[w]$, then the growth order
$\rho:=\rho(w)$ of $w(z)$ satisfies
$$
\rho \leq 1+\frac{\deg_{z,\infty} a}{nqk-W(P)},
$$
where $Q(z)$ is a polynomial of degree $q$.
\end{theorem}

 In 2011, Gu et al \cite{GLY} obtained Theorem \ref{thmB} when $k=1, q=1$.
In this article, we extend Theorem \ref{thmB}, and obtain the following
result.

\begin{theorem} \label{thm1}
 Let $k, m, n, q\in \mathbb{N}$, and let
$P[w]$ be a differential polynomial. If an entire function $w(z)$
whose all zeros have multiplicities at least $k$ satisfies the
differential equation $[([Q(w^{(k-1)}(z))]^n)']^m=P[w]$  and
$(nqk-nq+1)m> W(P)$, then
the growth order $\rho:=\rho(w)$ of $w(z)$ satisfies
$$
\rho\leq 1+\frac{\deg_{z,\infty}a}{(nqk-nq+1)m-W(P)},
$$
where  $Q(z)$ is a polynomial of degree $q$.
\end{theorem}

The following examples  show that the result is sharp in special
cases.

\begin{example}[\cite{s8}] \label{examp1}\rm
 For $n>0$, let
$w(z)=\cos{z^{\frac{n}{2}}}$, then $\rho(w)=\frac{n}{2}$ and $w$
satisfies the  algebraic differential equation:
$$
[(w^2)']^{2} = n^{2}z^{n-2}w^2(1-w^{2}) = 0.
$$
when $n=1$ or $2$,  $\deg_{z,\infty} a=0$, and the growth order
$\rho(w)$ of any entire solution $w(z)$ of \eqref{e1.3} satisfies
$\rho(w) \leq 1$; when $n \geq 3$, $\deg_{z,\infty} a=n-2$, and
the growth order $\rho(w)$ of any entire solution $w(z)$ of above
equation satisfies $\rho(w) \leq \frac{n}{2}$.
\end{example}

\begin{example} \label{examp2}\rm
For $n=2$, entire function $w(z)=e^{z^2}$
satisfies the  algebraic differential equation
$$
[(w')^2]'=8zw^2+8z^2w'w.
$$
We know $k=2$, $m=1$, $n=2$, $q=1$, $\deg_{z,\infty}a=2$, $W(P)=1$, and then
$\rho=2\leq 1+\frac{2}{3-1}=2$. This example illustrates that
Theorem \ref{thm1} is an extension result of Theorems  \ref{thmA} and  \ref{thmB}, and
our result is sharp in the special cases.
\end{example}
Set
\begin{equation}
\begin{gathered}
(Q(w_2^{(k)}(z)))^{m_1}=a(z)w_1^{(n)}\\
(w_1^{(n)})^{m_2}= P(w_2),
\end{gathered}\label{e1.3}
\end{equation}
where $m_1, m_2 \in \mathbb{N}$, $Q(z)$ and $a(z)$ are two
polynomials of degrees $q$ and $D(a)$, respectively.
In 2012, Qi et al proved the following theorem.

\begin{theorem}[\cite{QLY}] \label{thmC}
 Let $k, m_1, m_2, n, q\in \mathbb{N}$, and let $w=(w_1, w_2)$ be a pair of entire solutions of system \eqref{e1.3}, if ${m_1m_2}qk>W(P)$, and all zeros of $w_2$
have multiplicity at least $k$, then the growth orders $\rho(w_i)$
of $w_i(z)$ for $i=1,2$ satisfy
$$
\rho(w_1)= \rho(w_2)\leq 1+\frac{\deg_{z,\infty}a+D(a)}{{m_1m_2}qk-W(P)}.
$$
\end{theorem}

In 2009, Gu et al \cite{GDY} obtained Theorem \ref{thmC} when $k=1$, $q=1$
Now we consider the similar result to Theorem \ref{thm1} for the system of
the algebraic differential equations
\begin{equation}
\begin{gathered}[]
[([Q(w_2^{(k-1)})]^{m_3})']^{m_1}=a(z)R(w_1^{(n)}),\\
(R(w_1^{(n)}))^{m_2}=P[(w_2],
\end{gathered} \label{e1.4}
\end{equation}
where $R(z)$ is a polynomial, too.
We obtain the following result.

\begin{theorem} \label{thm2}
 Let $k, n, q, m_1,m_2,m_3\in \mathbb{N}$,
and let $w=(w_1, w_2)$ be a pair of entire solutions of
system \eqref{e1.4}. If $(m_3qk-m_3q+1){m_1m_2}>W(P)$, and all zeros of
$w_2$ have multiplicity at least $k$,  then the growth orders
$\rho(w_i)$ of $w_i(z)$ for $i=1,2$ satisfy
$$
\rho(w_1)=\rho(w_2)\leq 1
+\frac{\deg_{z,\infty}a+m_2 D(a)}{(m_3qk-m_3q+1){m_1m_2}-W(P)}.
$$
\end{theorem}

\begin{example} \label{examp3}
 The entire functions $w_1(z)=e^z+c$, $w_2(z)=e^z$ satisfy the
 algebraic differential equation system
\begin{gather*}
[(w^{(k-1)}_2)^2]'=2(w_1^{(n)})^2\\
\{(w_1^{(n)})^2\}^3 =(w_2)^3(w'_2)^2,
    \end{gather*}
where $c$ is a constant, $m_1=1$, $m_2=3$, $m_3=2$, $q=1$, $D(a)=0$,
$W(P)=2$, $\deg_{z,\infty}a=0$ and
$(m_3qk-m_3q+1){m_1m_2}=3(2k-1)>2=W(P)$. So
$\rho(w_1)=\rho(w_2)=1\leq 1$. So the conclusion of Theorem \ref{thm2} may
be applied.
\end{example}

\section{Main Lemmas}

To prove our result, we need the following lemmas. Lemma
\ref{lem1} extends the result by Zalcman \cite{s13} concerning normal family.


\begin{lemma}[\cite{s14}] \label{lem1}
Let $\mathcal{F}$ be a family of meromorphic(or analytic) functions on the unit disc.
 Then $\mathcal{F}$ is not normal on the unit disc if and only if there exist
\begin{itemize}
\item[(a)]  a  number $r\in (0,1)$;
\item[(b)]  points $z_n$  with $|z_n|<r$;
\item[(c)]  functions $f_n\in \mathcal{F}$;
\item[(d)]  positive  numbers $\rho_n\to 0$
\end{itemize}
such that  $g_n(\zeta):=f_n(z_n+\rho_n\zeta)$ converges locally
uniformly to a nonconstant meromorphic (or entire) function
$g(\zeta)$, its order is at most 2. In particular, we may choose
$w_n$ and $\rho_n$, such that
$$
\rho_n\leq \frac{2}{f_n^{\sharp}(w_n)},
 f_n^{\sharp}(w_n)\geq f_n^{\sharp}(0).
$$ 
\end{lemma}

\begin{lemma}[\cite{s5}] \label{lem2}
 Let $f(z)$ be holomorphic  in the complex plane, $\sigma >-1$. If
$f^{\sharp}(z)= O(r^{\sigma})$,
then
$T(r,f)=O(r^{\sigma+1})$.
\end{lemma}

\begin{lemma}[\cite{GLY}] \label{lem3}
 Let  $f(z)$ be holomorphic in whole complex plane with growth order $\rho:=\rho(f)>1$, then
for each $0<\mu < \rho -1,$ there exists a sequence
$a_n\to \infty,$ such that
\begin{equation}
\lim_{n\to
\infty}\frac{f^{\sharp}(a_n)}{|a_n|^{\mu}}=+\infty. \label{e2.1}
\end{equation}
\end{lemma}

\section{Proofs of theorems}

\begin{proof}[Proof of Theorem \ref{thm1}]
Suppose that the conclusion of theorem is not true, then there exists 
an entire solution $w(z)$ satisfies the equation $[([Q(w^{(k-1)}(z))]^n)']^m=P[w]$,
such that
\begin{equation}
\rho> 1+\frac{\deg_{z,\infty}a}{(nqk-nq+1)m-W(P)}. \label{e3.1}
\end{equation}
By Lemma \ref{lem2} we know that for each $0<\mu<\rho-1$, there exists a
sequence of points $a_j\to\infty(j\to\infty)$,
such that \eqref{e2.1} is valid. This implies that the family
$\{w_j(z):=w(a_j+z)\}_{j\in \mathbb{N}}$ is not normal at $z=0$.
By Lemma \ref{lem1}, there exist sequences $\{b_j\}$ and $\{\rho_j\}$
such that
\begin{equation}
|a_j-b_j|<1, \quad \rho_j\to 0,\label{e3.2}
\end{equation}
and $g_j(\zeta):=w_j(b_j-a_j+\rho_j\zeta)=w(b_j+\rho_j\zeta)$
converges locally uniformly to a nonconstant entire function
$g(\zeta)$, which order is at most 2, all zeros of $g(\zeta)$ have
multiplicity at least $k$ . In particular, we may choose $b_j$ and
$\rho_j$, such that
\begin{equation}
\rho_j\leq\frac{2}{w^{\sharp}(b_j)},\quad  w^{\sharp}(b_j)\geq
w^{\sharp}(a_j).\label{e3.3}
\end{equation}
According to \eqref{e2.1} and \eqref{e3.1}--\eqref{e3.3}, we can get the following
conclusion:
For any fixed constant $0\leq\mu<\rho-1$, we have
\begin{equation}
\lim_{j\to\infty}b_j^{\mu}\rho_j=0.\label{e3.4}
\end{equation}

In the differential equation $[([Q(w^{(k-1)}(z))]^n)']^m=P[w]$, we
now replace $z$ by $b_j+\rho_j\zeta$. Assuming that $P[w]$ has the
form \eqref{e1.2}. Then we obtain
$$
[([Q(w^{(k-1)}(b_j+\rho_j\zeta))]^n)']^m=\sum_{r\in I}
a_r(b_j+\rho_j\zeta, g_j(\zeta))\rho_j^{-p(r)}M_{r}[g_j](\zeta).
$$
From
\begin{align*}
&([Q(w^{(k-1)}(b_j+\rho_j\zeta))]^n)'\\
&=
n[Q(w^{(k-1)}(b_j+\rho_j\zeta))]^{n-1}Q'(w^{(k-1)}(b_j+\rho_j\zeta))
(w^{(k)}(b_j+\rho_j\zeta)),
\end{align*}
we have
\begin{align*}
&([Q(w^{(k-1)}(b_j+\rho_j\zeta))]^n)'\\
&=\rho_j^{-(nqk-nq+1)}g_j^{(k)}(\zeta)[nq(g_j^{(k-1)})^{(nq-1)(k-1)}
(\zeta)+H(\rho_j,g_j^{(k-1)}(\zeta))],
\end{align*}
where $H(s,t)$ is a polynomial in two variables, whose degree
$\deg_{s}H$ in $s$ satisfies $\deg_{s}H\geq 1$.
Hence we deduce that
\begin{align*}
&\rho_j^{-(nqk-nq+1)m}\{g_j^{(k)}(\zeta)[nq(g_j^{(k-1)})^{(nq-1)(k-1)}(\zeta)
+H(\rho_j,g_j^{(k-1)}(\zeta))]\}^m \\
&=\sum_{r\in I} a_r(b_j+\rho_j\zeta,
g_j(\zeta))\rho_j^{-p(r)}M_{r}[g_j](\zeta).
\end{align*}
Therefore,
\begin{equation}
\begin{aligned}
&\{g_j^{(k)}(\zeta)[nq(g_j^{(k-1)})^{(nq-1)(k-1)}(\zeta)
 +H(\rho_j,g_j^{(k-1)}(\zeta))]\}^m\\
&=\sum_{r\in I}\frac{ a_r(b_j+\rho_j\zeta, g_j(\zeta))}{b_j^{\deg
a_r}}[b_j^{\frac{\deg a_r}{(nqk-nq+1)m-p(r)}}\rho_j]^{(nqk-nq+1)m-p(r)}M_{r}[g_j](\zeta).
\end{aligned}
\label{e3.5}
\end{equation}
Because
$0\leq\mu=\frac{\deg_{z,\infty}a_r}{(nqk-nq+1)m-p(r)}
\leq\frac{\deg_{z,\infty}a}{(nqk-nq+1)m-W(P)}<\rho-1$,
$p(r)<(nqk-nq+1)m$, for every fixed $\zeta\in\mathrm{C}$, if
$\zeta$ is not the zero of $g(\zeta)$, by \eqref{e3.4} then we can get
$g^{(k)}(\zeta)=0$ from \eqref{e3.5}. By the all zeros of $g(\zeta)$ have
multiplicity at least $k$, this is a contradiction.
The proof is complete.
\end{proof}



\begin{proof}[Proof of Theorem \ref{thm2}]
By the first equation of the systems
of algebraic differential equations \eqref{e1.4}, we know
$$
R(w_1^{(n)})=\frac{[([Q(w_2^{(k-1)})]^{m_3})']^{m_1}}{a(z)}.
$$
Therefore,  
$\rho(w_1)=\rho(w_2)$.

If $w_2$ is a rational function, then $w_1$ must be a rational
function, so that the conclusion of Theorem \ref{thm2} is right. If $w_2$
is a transcendental entire function, by the systems of algebraic
differential equations \eqref{e1.4}, then we have
\begin{equation}
[([Q(w_2^{(k-1)})]^{m_3})']^{m_1m_2}=(a(z))^{m_2}P[w_2].\label{e3.6}
\end{equation}

Suppose that the conclusion of Theorem \ref{thm2} is not true, then there
exists an entire vector $w(z)=(w_1(z),w_2(z))$ which satisfies the
system of equations \eqref{e1.4} such that
\begin{equation}
\rho:=\rho(w_2)> 1+\frac{\deg_{z,\infty}a+
m_2D(a)}{(m_3qk-m_3q+1)m_1m_2-W(P)},\qquad\quad. \label{e3.7}
\end{equation}
By Lemma \ref{lem2} we know that for each $0<\mu<\rho-1$, there exists a
sequence of points $a_j\to\infty(j\to\infty)$,
such that \eqref{e2.1} is right. This implies that the family
$\{w_j(z):=w(a_j+z)\}_{j\in \mathbb{N}}$ is not normal at $z=0$.
By Lemma \ref{lem1}, there exist sequences $\{b_j\}$ and $\{\rho_j\}$
such that
\begin{equation}
|a_j-b_j|<1, \quad \rho_j\to 0,\label{e3.8}
\end{equation}
and
$g_j(\zeta):=w_{2,j}(b_j-a_j+\rho_j\zeta)=w_2(b_j+\rho_j\zeta)$
converges locally uniformly to a nonconstant entire function
$g(\zeta)$, which order is at most 2, all zeros of $g(\zeta)$ have
multiplicity at least $k$ . In particular, we may choose $b_j$ and
$\rho_j$, such that
\begin{equation}
\rho_j\leq\frac{2}{w^{\sharp}_2(b_j)},\quad  w_2^{\sharp}(b_j)\geq
w_2^{\sharp}(a_j).\label{e3.9}
\end{equation}
According to \eqref{e3.6} and \eqref{e3.7}--\eqref{e3.9}, we can get the following
conclusion:
For any fixed constant $0\leq\mu<\rho-1$, we have
\begin{equation}
\lim_{m\to\infty}b_j^{\mu}\rho_j=0.\label{e3.10}
\end{equation}

In the differential equation \eqref{e3.6} we now replace $z$ by
$b_j+\rho_j\zeta$, then we obtain
\begin{align*}
&[([Q(w^{(k-1)}(b_j+\rho_j\zeta))]^{m_3})']^{{m_1m_2}}\\
&=(a(b_j+\rho_j\zeta))^{m_2}
\sum_{r\in I} a_r(b_j+\rho_j\zeta,
g_j(\zeta))\rho_j^{-p(r)}M_{r}[g_j](\zeta).
\end{align*}
By
\begin{align*}
&([Q(w^{(k-1)}(b_j+\rho_j\zeta))]^{m_3})'\\
&={m_3}[Q(w^{(k-1)}(b_j+\rho_j\zeta))]^{{m_3}-1}Q'(w^{(k-1)}(b_j+\rho_j\zeta))
(w^{(k)}(b_j+\rho_j\zeta)),
\end{align*}
we obtain
\begin{align*}
&([Q(w^{(k-1)}(b_j+\rho_j\zeta))]^{m_3})'\\
&=\rho_j^{-({m_3}qk-m_3q+1)}g_j^{(k)}(\zeta)[{m_3}q(g_j^{(k-1)})^{({m_3}q-1)(k-1)}
(\zeta)+H(\rho_j,g_j^{(k-1)}(\zeta))],
\end{align*}
where $H(s,t)$ is a polynomial in two variables, whose degree
$\deg_{s}H$ in $s$ satisfies $\deg_{s}H\geq 1$.
Therefore,
\begin{align*}
&\rho_j^{-(m_3qk-m_3q+1){m_1m_2}}\{g_j^{(k)}(\zeta)
[m_3q(g_j^{(k-1)})^{(m_3q-1)(k-1)}(\zeta)+
H(\rho_j,g_j^{(k-1)}(\zeta))]\}^{{m_1m_2}} \\
&=\sum_{r\in I}
(a(b_j+\rho_j\zeta))^{m_2}a_r(b_j+\rho_j\zeta,
g_j(\zeta))\rho_j^{-p(r)}M_{r}[g_j](\zeta).
\end{align*}
Thus
\begin{equation}
\begin{aligned}
&\{g_j^{(k)}(\zeta)[m_3q(g_j^{(k-1)})^{(m_3q-1)(k-1)}(\zeta)+H(\rho_j,g_j^{(k-1)}(\zeta))]\}^{{m_1m_2}}\\
&=\sum_{r\in I}\frac{ (a(b_j+\rho_j\zeta))^{m_2}a_r(b_j+\rho_j\zeta, g_j(\zeta))}{b_j^{\deg
a_r+m_2D(a)}}\\
&\quad \times[b_j^{\frac{\deg a_r+m_2D(a)}{(m_3qk-m_3q+1){m_1m_2}-p(r)}}
\rho_j]^{(m_3qk-m_3q+1){m_1m_2}-p(r)}M_{r}[g_j](\zeta).
\end{aligned} \label{e3.11}
\end{equation}
For every fixed $\zeta\in\mathrm{C}$, if $\zeta$ is not a zero of
$g(\zeta)$, for $j\to\infty$ and
\begin{align*}
0&\leq\mu =\frac{\deg_{z, \infty}a_r+m_2D(a)}{(m_3qk-m_3q+1){m_1m_2}-p(r)}\\
&\leq\frac{\deg_{z,\infty}a+m_2D(a)}{(m_3qk-m_3q+1){m_1m_2}
-\deg P(w_2)}<\rho-1
\end{align*}
then we have $(g^{(k)})^{{m_1m_2}}=0$, which
contradicts with all zeros of $g(\zeta)$ have multiplicity at
least $k$. So
\[
\rho(w_2)\leq 1+\frac{\deg_{z,\infty}a+m_2D(a)}{(m_3qk-m_3q+1){m_1m_2}-\deg
P(w_2)}.
\]
\end{proof}



\subsection*{Acknowledgments}
 This work was supported by the Visiting Scholar
Program of Chern Institute of Mathematics at Nankai University.
The first author would like to express his hearty thanks to Chern
Institute of Mathematics provided very comfortable research
environments to him worked as visiting scholar. The authors
finally wish to thank Professor Barsegian for his helpful comments
and suggestions.


\begin{thebibliography} {00}

\bibitem{s1}  Bank, S.;  Kaufman, R.;
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