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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 269, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/269\hfil Value distribution]
{Value distribution of difference polynomials of meromorphic functions}

\author[Y. Liu, X. Qi, H. Yi \hfil EJDE-2013/269\hfilneg]
{Yong Liu, Xiaoguang Qi, Hongxun Yi}  % in alphabetical order

\address{Yong Liu \newline
 Department of Mathematics, Shaoxing College of Arts and Sciences,
Shaoxing, Zhejiang 312000,  China.\newline
Department of Physics and Mathematics, Joensuu Campus,
University of Eastern Finland, P.O. Box 111, Joensuu FI-80101, Finland}
\email{liuyongsdu@aliyun.com}

\address{Xiaoguang Qi \newline
Department of Mathematics, Jinan University, 250022  Jinan Shandong,
China}
\email{xiaoguangqi@yahoo.cn}

\address{Hongxun Yi \newline
School of Mathematics, Shandong University, 250100 Jinan Shandong, China}
\email{hxyi@sdu.edu.cn}

\thanks{Submitted June 18, 2013. Published December 5, 2013.}
\subjclass[2000]{30D35, 39B12}
\keywords{Meromorphic function; difference polynomial; uniqueness; finite order}

\begin{abstract}
 In this article, we study the value distribution of difference
 polynomials of meromorphic functions, and obtain some results which
 can be viewed as discrete analogues of  the results given by Yi and
 Yang \cite{y2}.  We also consider the value distribution of
 $$
 \varphi(z)=f(z)(f(z)-1)\prod_{j=1}^{n}f(z+c_j).
 $$
\end{abstract}

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

\section{Introduction and main results}

In this article, we  assume that the reader is familiar with the fundamental 
results and the standard notation of the Nevanlinna theory 
(see, e.g.,  \cite{h5,y1}). 
Let $f(z)$ and $g(z)$ be two non-constant meromorphic functions in
the complex plane. By $S(r,f)$, we denote any quantity satisfying
$S(r,f)=o(T(r,f))$ as $r \to \infty$, possibly outside a set
of   finite logarithmic measure. Then the meromorphic function
$\alpha$ is called a small function of   $f(z)$, if $T(r,\alpha)=S(r,f)$.
If $f(z)-\alpha$ and $g(z)-\alpha$ have same zeros, 
counting multiplicity (ignoring multiplicity), then we say $f(z)$ and $g(z)$
share the small function $\alpha$ CM  (IM).  Denote
$$
\delta(\alpha, f)= \liminf_{r\to\infty}\frac{m(r, \frac{1}{f-\alpha})}{T(r, f)},
$$
where $\alpha$ is a small function related to $f(z)$. 

In the following sections, we denote by $E$ a set
of finite logarithmic  measure,  it is not necessarily the same 
at each appearance.
In 1991, Yi and Yang \cite{y2}  obtained the following theorem.


\begin{theorem} \label{thmA}
 Let $f(z)$ and $g(z)$ be  meromorphic functions satisfying 
$\delta(\infty, f)=\delta(\infty, g)=1$.
If $f'$ and $g'$ share $1$ CM and $\delta(0, f)+\delta(0, g)>1$,
then either $f\equiv g$ or $f'g'\equiv 1$.
\end{theorem}

Lately, there has been an increasing interest in studying difference 
equations in the complex plane. For example, Halburd and Korhonen 
\cite{h1,h2} established a version of Nevanlinna theory based on difference operators. 
Ishizaki and Yanagihara \cite{i1} developed a version of Wiman-Valiron theory 
for difference equations of entire functions of small growth. 
Also Chiang and Feng \cite{c1} has a difference version of  Wiman-Valiron.
The main purpose  of this article is to establish  partial difference 
counterparts of Theorem \ref{thmA}.  Our results can be stated as follows.

\begin{theorem} \label{thm1.1}
Let $c_j, a_j, b_j (j=1,2,\dots,k)$ be complex constants, and  
let $f(z)$ and $g(z)$ be two nonconstant finite order  meromorphic functions 
satisfying $\delta(\infty, f)=\delta(\infty, g)=1$. 
Let $L(f)=\sum _{i=1}^{k}a_if(z+c_i)$ and $ L(g)=\sum _{i=1}^{k}b_ig(z+c_i)$. 
Suppose that $L(f) \cdot L(g)\not \equiv 0$. If
 $L(f)$ and $ L(g)$ share $1$ CM and $\delta(0, f)+\delta(0, g)>1$, then
$L(f)L(g)\equiv 1$ or $L(f)\equiv L(g)$.
\end{theorem}

\begin{theorem} \label{thm1.2}
Suppose that $f(z)$ is a nonconstant meromorphic function.
Let $\delta_f=\sum_{a\in \mathbb{C}}\delta(a, f)$. If 
$\Delta_cf(z)=f(z+c)-f(z)\not\equiv 0$ $(c\in\mathbb{C}\setminus \{0\})$,  then
$$
N\Big(r, \frac{1}{\Delta_cf(z)}\Big)
\leq \Big((1-\frac{\delta_f}{2}+\varepsilon)T(r, \Delta_cf(z)\Big)\quad
 (r\not\in E),
$$
where $\varepsilon$ is any fixed positive number.
\end{theorem}

Recently, Zhang \cite{z1} considered the value distribution of difference 
polynomial of entire functions, and obtain the following result.

\begin{theorem} \label{thmB}
Let $f(z)$ be a transcendental entire function of finite order, and $\alpha(z)$
be a small function with respect to $f(z)$. Suppose that $c$ is a non-zero 
complex constant and $n$ is an integer.
If $n\geq 2$, then $f^{n}(z)(f(z)-1)f(z+c)-\alpha(z)$ has infinitely many zeros.
\end{theorem}

A natural question arises: If $n=1$, whether we can get a similar  
conclusion? The following theorems give a partial answer to this
question.

\begin{theorem} \label{thm1.3}
 Let $f(z)$ be a finite order transcendental entire function with one Borel 
exceptional value $d$ according to the condition or its proof described.
Let $c_j(j=1,\dots, n), b$ be complex constants. If $d^{n+2}-d^{n+1}-b\neq 0$, 
then  $\varphi(z)=f(z)(f(z)-1)\prod_{j=1}^{n}f(z+c_j)-b$ has infinitely
many zeros and $\lambda(\varphi(z)-b)=\rho(f)$.
\end{theorem}

\begin{theorem} \label{thm1.4}
 Let $f(z)$ be a finite order transcendental entire function, and let 
$c_j$ $(j=1,\dots, n)$, $b_j$ $(j=1,\dots, n), b$ be complex constants. 
If $f(z)$ or $f(z)-1$ has infinitely many multi-order zeros, then 
$f(z)(f(z)-1)\prod_{i=1}^{n}(f(z+c_i)-b_i)-b$  has infinitely many zeros.
\end{theorem}

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

We need the following lemmas.
The first lemma is a difference analogue of the logarithmic derivative lemma.

\begin{lemma}[\cite{h1}] \label{lem2.1}
Let $f(z)$ be a meromorphic function of finite order  and let $c$ be
a non-zero complex number. Then for any small periodic  function $a(z)$ 
with period $c$,
$$
m\Big(r, \frac{f(z+c)-f(z)}{f(z)-a(z)}\Big)=S(r, f).
$$
\end{lemma}

The following lemma is essential for our proof and is due to
Heittokangas et al., see \cite[Theorems 6 and 7]{h6}.

\begin{lemma}[\cite{h3}] \label{lem2.2}
Let $f(z)$ be a meromorphic function of finite order, $c\neq 0$ be fixed. Then
\begin{gather*}
\overline{N}(r, f(z+c))\leq \overline{N}(r, f(z))+S(r, f), \\
N(r, f(z+c))\leq N(r, f(z))+S(r, f).
\end{gather*}
\end{lemma}

\begin{lemma} \label{lem2.3}
Let $f$ be a nonconstant meromorphic function of finite order such 
that $\delta(\infty, f)=1$ and $\delta(0, f)>0$. Let $L(f)$ be 
as in Theorem \ref{thm1.1}. Then
$$
T(r, f)\leq \Big(\frac{1}{\delta(0, f)}+\varepsilon\Big)T(r, L(f)),\quad
 r\not\in E,
$$
and
$$
N\Big(r, \frac{1}{L(f)}\Big)<(1-\delta(0, f)+\varepsilon+o(1))T(r, L(f)),
\quad r\not\in E,
$$
where $\varepsilon>0$ can be fixed arbitrarily.
\end{lemma}

\begin{proof} 
From $\delta(\infty, f)=1$, we have
$$
N(r, f)=o(T(r, f)).
$$
Then from  Lemmas \ref{lem2.1} and \ref{lem2.2}, we obtain
\begin{equation}
\begin{aligned}
T(r, L(f))&=m(r, L(f))+N(r, L(f))\\
 &\leq m(r, f)+m\Big(r, \frac{L(f)}{f}\Big)+k N(r, f)+o(T(r, f))\\
 &\leq (1+o(1))T(r, f)),\;\;r\not\in E.
\end{aligned}\label{e2.1}
\end{equation}
On the other hand,
\begin{equation}
\begin{aligned}
m\Big(r, \frac{1}{f}\Big)
&\leq m\Big(r, \frac{1}{L(f)}\Big)+m\Big(r, \frac{L(f)}{f}\Big)\\
&= m\Big(r, \frac{1}{L(f)}\Big)+o(T(r, f))\\
&= T(r, L(f))-N\Big(r, \frac{1}{L(f)}\Big)+o(T(r, f)),\quad
 r\not\in E. \label{e2.2}
\end{aligned}
\end{equation}
By the definition of $\delta(0, f)$, we obtain
\begin{equation}
m\Big(r, \frac{1}{f}\Big)\geq (\delta(0, f)-\varepsilon)T(r, f),\label{e2.3}
\end{equation}
where $\varepsilon>0$ can be fixed arbitrarily.
Combining \eqref{e2.2} and \eqref{e2.3} yields
$$
(\delta(0, f)-\varepsilon)T(r, f)<T(r, L(f)), \quad r\not\in E;
$$
that is,
$$
T(r, f)<\Big(\frac{1}{\delta(0, f)}+\varepsilon\Big)T(r, L(f)),\quad r\not\in E.
$$
By \eqref{e2.1}, \eqref{e2.2} and \eqref{e2.3}, we have
$$
(\delta(0, f)-\varepsilon+o(1))T(r, L(f))<T(r, L(f))
-N\Big(r, \frac{1}{L(f)}\Big),\quad r\not\in E;
$$
that is,
$$
N\Big(r, \frac{1}{L(f)}\Big)<(1-\delta(0, f)+\varepsilon+o(1))T(r, L(f)),\quad
r\not\in E.
$$
\end{proof}

\begin{lemma}[\cite{y2}] \label{lem2.4}
Let $f_1, f_2$ and $f_3$ be three meromorphic  functions satisfying
$$
\sum_{i=1}^{3}f_i\equiv 1.
$$
Assume that  $f_1$ is not constant, and
$$
\sum_{i=1}^{3}N_2\big(r, \frac{1}{f_i}\big)+\sum_{i=1}^{3}N(r, f_j)
\leq (\lambda+o(1))T(r)\quad  (r\in I),
$$
where $\lambda<1$, $T(r)=\max\{T(r,f_i)|i=1,2,3)\}$,  $N_2(r, 1/f_j)$ 
is the counting function of zeros of $f_j(j=1,2,3)$, where a multiple 
zero is counted two times, and a simple zero is counted once.  
Then $f_2\equiv 1$ or $f_3\equiv 1$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
Set $I_1=\{r:T(r, L(f))\geq T(r, L(g))\}\subseteq (0, \infty)$  and 
$I_2=(0, \infty)\backslash I_1$. Then there is at least one
$I_i$ $(i=1,2)$ such that $I_i$ has infinite logarithmic measure.
 Without loss of generality, we may suppose that $I_1$ has infinite
 logarithmic measure.

Because $\delta(0, f)+\delta(0, g)>1$, it follows that $\delta(0, f)>0$ 
and $\delta(0, g)>0$. Lemma \ref{lem2.3} yields 
\begin{gather*}
T(r, f)=O(T(r, L(f))), \quad r\in I_1\backslash E, \\
T(r, g)=O(T(r, L(g)))=O(T(r, L(f))),\;\;\;r\in I_1\backslash E.
\end{gather*}
Thus,
\begin{gather*}
N(r, f)=o(T(r, L(f))),\quad r\in I_1\backslash E, \\
N(r, g)=o(T(r, L(f))),\quad r\in I_1\backslash E.
\end{gather*}
Since $L(f)=\sum _{i=1}^{k}a_if(z+c_i)$ and $L(g)=\sum _{i=1}^{k}b_ig(z+c_i)$
 share $1$ CM, we have
\begin{equation}
\frac{L(f)-1}{L(g)-1}=h(z), \label{e2.4}
\end{equation}
where
$$
N(r, h)+N(r, \frac{1}{h})\leq kN(r, f)+kN(r, g)+o(T(r, f))+o(T(r, g))
=o(T(r, L(f))),
$$
for $r\in I_1\backslash E$.
Let $f_1=L(f)$, $f_2=h(z)$,  $f_3=-L(g)h(z)$.  Then we obtain
$f_1+f_2+f_3\equiv1$,
and
$$
\sum_{i=1}^{3}N(r, f_i)\leq kN(r, f)+kN(r, g)+2N(r, h)=o(T(r)),\quad
r\in I_1\backslash E,
$$
where $T(r)=\max_{1\leq i\leq 3}\{T(r, f_j)\}$. For any 
$\varepsilon$ satisfying $0<\varepsilon<\frac{\delta(0, f)+\delta(0, g)-1}{4}$,
by Lemma \ref{lem2.3}, we obtain
\begin{align*}
\sum_{i=1}^{3} N_2\Big(r, \frac{1}{f_j}\Big)
&\leq N\Big(r, \frac{1}{L(f)}\Big)+N\Big(r, \frac{1}{L(g)}\Big)+2N(r, h)\\
&\leq (2-\delta(0, f)-\delta(0, g)+o(1)+2\varepsilon)T(r)\\
&=(\lambda+o(1))T(r),\quad r\in I_1\backslash E,
\end{align*}
where $\lambda=2-\delta(0, f)-\delta(0, g)+2\varepsilon<1$.
If $f_1(z)=L(f)$ is a constant, by Lemma \ref{lem2.3}, we see that  $f(z)$
is also a constant, a contradiction. Hence, $f_1(z)$ is not constant.
By Lemma \ref{lem2.4}, we obtain
$f_2\equiv 1$  or $f_3\equiv 1$.
If $f_2\equiv 1$, then we obtain  $L(f)\equiv L(g)$.
If $f_3\equiv 1$, we have $L(f)\equiv-\frac{1}{h}$, $L(g)\equiv -h$,
 and so $L(f)L(g)\equiv 1$.
\end{proof}

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

We need the following lemmas.

\begin{lemma}[\cite{h3,h4}] \label{lem3.1}
 Let $f(z)$ be a nonconstant finite order meromorphic function and let 
$c\neq 0$ be an arbitrary complex number. Then
$$
T(r, f(z+|c|))=T(r, f(z))+S(r, f).
$$
\end{lemma}

\begin{remark} \label{rmk1}\rm   
It is shown in  \cite[p. 66]{g1}, that for $c\in \mathbb{C}\setminus \{0\}$,
 we have
$$
(1+o(1))T(r-|c|, f(z))\leq T(r, f(z+c))\leq (1+o(1))T(r+|c|, f(z))
$$
hold as $r\to \infty$, for a general meromorphic function. 
By this and Lemma \ref{lem3.1}, we obtain
 $$
T(r, f(z+c))=T(r, f(z))+S(r, f)
$$
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
Without loss of generality, we assume that there exist infinitely many 
values $a$ such that $\delta(a, f)>0$. Then there is an sequence
$\{a_i\}_{i=1}^{\infty}$ satisfying $a_i\neq a_j (i\neq j)$ and 
$\sum_{i=1}^{\infty}\delta(a_i, f)=\delta_f$. Hence for any fixed 
positive number $\varepsilon$, there exists an integer $q$ such that
\begin{equation}
\delta=\sum_{i=1}^{q}\delta(a_i, f)>\delta_f-\frac{\varepsilon}{3}.\label{e3.1}
\end{equation}
Set
$$
F(z)=\sum_{i=1}^{q}\frac{1}{f(z)-a_i}.
$$
Then
\begin{equation}
\sum_{i=1}^{q}m\Big(r, \frac{1}{f-a_i}\Big)=m(r, F)+O(1).\label{e3.2}
\end{equation}
Hence by Lemma \ref{lem2.1}, we obtain
\begin{equation}
\begin{aligned}
m(r, F)
&\leq m\Big(r, \frac{1}{\Delta_cf(z)}\Big)+\sum_{i=1}^{q}m
\Big(r, \frac{\Delta_cf(z)}{f-a_i}\Big)+O(1)\\
&=T(r, \Delta_cf(z))-N\Big(r, \frac{1}{\Delta_cf(z)}\Big)+S(r, f).\label{e3.3}
\end{aligned}
\end{equation}
and
\begin{equation}
\sum_{i=1}^{q}m\Big(r, \frac{1}{f-a_i}\Big)
\geq \Big(\delta_f-\frac{\varepsilon}{3}\Big)T(r, f),\quad
r\not\in E.\label{e3.4}
\end{equation}
From \eqref{e3.1}-\eqref{e3.4} and Remark 1, we have
\begin{align*}
N(r, \frac{1}{\Delta_cf(z)})
&\leq T(r, \Delta_cf(z))-(\delta_f-\frac{2}{3}\varepsilon)T(r, f)\\
&\leq T(r, \Delta_cf(z))-(\frac{\delta_f}{2}-\varepsilon)T(r, \Delta_cf(z))\\
&= (1-\frac{\delta_f}{2}+\varepsilon)T(r, \Delta_cf(z)), \quad r\not\in E.
\end{align*}
\end{proof}

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

The following lemma is a generalization  of Borel's Theorem on  
linear combinations of  entire functions.

\begin{lemma}[{\cite[pp.  79-80]{y1}}] \label{lem4.1}
 Let $f_j(z)$ $(j=1,2,\dots,n; n\geq 2)$ be meromorphic functions, 
$g_j(z)$ $(j=1,2,\dots,n)$ be entire functions, and assume they satisfy
\begin{itemize}
\item[(i)] $f_1(z)e^{g_1(z)}+\dots+ f_k(z)e^{g_k(z)}\equiv 0$;

\item[(ii)] when $1\leq j<k\leq n$, then $g_j(z)-g_k(z)$ is not a constant.

\item[(iii)] when $1\leq j\leq n$, $1 \leq h<k \leq n$, then
$$
T(r, f_j)=o\{T(r, e^{g_h-g_k})\}\quad (r\to \infty,\;r\not\in E_1),
$$
where $E\subset (1, \infty)$ is of  finite logarithmic measure.
\end{itemize}
Then $f_j\equiv 0$ $(j=1,\dots,n)$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm1.3}]

Set $\varphi(z)=f(z)(f(z)-1)\prod_{j=1}^{n}f(z+c_j)$. 
Next we prove that $\rho(\varphi)=\rho(f)$. We write $\varphi(z)$ as 
\begin{equation}
\varphi(z)=f^{n+1}(z)(f(z)-1)\prod_{j=1}^{n}
\Big(\frac{f(z+c_j)}{f(z)}\Big)\label{e4.1}
\end{equation}
By Lemma \ref{lem2.1}, we obtain
\begin{equation}
\begin{aligned}
T(r, \varphi)=m(r, \varphi)
&\leq (n+1)m(r, f)+m(r, f-1)+\sum_{i=1}^{n} m
 \Big(r, \frac{f(z+c_j)}{f(z)}\Big)+S(r, f)\\
&=(n+2)T(r, f)+S(r, f). 
\end{aligned} \label{e4.2}
\end{equation}
On the other hand,
\begin{equation}
\begin{aligned}
(n+2)T(r, f)&=(n+2)m(r, f)\\
&=m(r, f^{n+1}(z)(f(z)-1))+S(r, f)\\
&\leq m(r, \varphi)+\sum_{i=1}^{n}m\Big(r, \frac{f(z)}{f(z+c_i)}\Big)+S(r, f)\\
&=T(r, \varphi)+S(r, f). \label{e4.3}
\end{aligned}
\end{equation}
By \eqref{e4.2} and \eqref{e4.3}, we have
\begin{equation}
\rho(\varphi)=\sigma(f).\label{e4.4}
\end{equation}
Suppose that $d$ is the Borel exceptional value of $f(z)$. Then we
can write $f(z)$ as
\begin{equation}
f(z)=d+g(z)\exp\{\alpha z^{k}\},\label{e4.5}
\end{equation}
where $\alpha$ is a nonzero constant, $k\geq 1$ is an integer, and $g(z)$
is an entire function such that $g(z)(\not\equiv0), \rho(g)<k$.
By \eqref{e4.5}, we have
\begin{equation}
f(z+c_j)=d+g(z+c_j)g_j(z)\exp \{\alpha z^{k}\},(j=1,2,\dots,n),\label{e4.6}
\end{equation}
where $g_j(z)=\exp\{\alpha(_{1}^{k})z^{k-1}c_j+\alpha(_{2}^{k})z^{k-2}c_j^{2}
+\dots+\alpha c_j^{k}\}$,  $\rho(g_j)=k-1$. Equality \eqref{e4.4} implies
that $\rho(\varphi-b)=\rho(f)$.
Next, we prove that $\lambda(\varphi(z)-b)=\rho(f)$. Suppose, contrary
to the assertion, that $\lambda(\varphi(z)-b)<\rho(f)$. Then
\begin{equation}
\varphi(z)-b=u(z)\exp\{\beta z^{k}\},\label{e4.7}
\end{equation}
where $u(z)$ is an entire  function with
$\rho(u)\leq \max\{\lambda(\varphi(z)-b), k-1\}<k$, and $\beta$ is
a nonzero constant. From \eqref{e4.5}-\eqref{e4.7},
we have
\begin{equation}
\begin{aligned}
&g^{2}(z)\prod_{i=1}^{n}g(z+c_i)g_i(z)\exp\{(n+2)\alpha z^{k}\}
+G_{n+1}(z)\exp\{(n+1)\alpha z^{k}\}\\
&+\dots+G_1(z)\exp\{\alpha z^{k}\}+d^{n+2}-d^{n+1}-b
=u(z)\exp\{\beta z^{k}\}, \label{e4.8}
\end{aligned}
\end{equation}
where $G_i(z)$ $(i=1, \dots, n+1)$ are difference polynomials in
$g(z)$, $g_1(z)$, $g_2(z)$, \dots, $g_n(z)$, $g_1(z+c_1)$, $g_2(z+c_2)$, 
\dots, $g_{n}(z+c_n)$. Since $g(z)\not\equiv 0$, by comparing the 
growth of both side of \eqref{e4.8}, we have $\beta=(n+2)\alpha$. 
Hence we can rewritten \eqref{e4.8} as
\begin{equation}
\begin{aligned}
&(g^{2}(z)\prod_{i=1}^{n}g(z+c_i)g_i(z)-u(z))
 \exp\{(n+2)\alpha z^{k}\}+G_{n+1}(z)\exp\{(n+1)\alpha z^{k}\}\\
&+\dots+G_1(z)\exp\{\alpha z^{k}\}+d^{n+2}-d^{n+1}-b=0. 
\end{aligned} \label{e4.9}
\end{equation}
By Lemma \ref{lem4.1} and \eqref{e4.9},  we obtain that $d^{n+2}-d^{n+1}-b=0$,
a contradiction. Hence, we obtain $\lambda(\varphi(z)-b)=\rho(f)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.4}]
Suppose that  $f(z)$ or $f(z)-1$ has infinitely many multi-order zeros.
 If $b=0$, then $H(z)$ has infinitely many zeros. Next we suppose that 
$b\neq 0$. If $H(z)-b$ has only finitely many zeros, then $H(z)$ can 
be rewritten as
\begin{equation}
H(z)=f(z)(f(z)-1)\prod_{i=1}^{n}(f(z+c_i)-b_i)-b=p(z)e^{q(z)},\label{e5.1}
\end{equation}
where $p(z), q(z)$ are polynomials.  Suppose that $H(z)$ is a polynomial.
Then we have
\begin{equation}
H(z)=f(z)(f(z)-1)\prod_{i=1}^{n}(f(z+c_i)-b_i)-b=P(z),\label{e5.2}
\end{equation}
where $P(z)$ is a polynomial. From \eqref{e5.2}, we have
\begin{align*}
(n+2)T(r, f)&=T(r, H(z))+S(r, f)\\
&=T(r, f(z)(f(z)-1)\prod_{i=1}^{n}(f(z+c_i)-b_i)-b)+S(r, f)\\
&=T(r, P(z))+S(r, f)=O(\log r)+S(r, f).
\end{align*}
This is impossible, since $f(z)$ is transcendental.
Hence $H(z)$ is transcendental, so we get
$p(z)\not \equiv 0$, $\deg q(z)\geq 1 $, by this, we  obtain
$p'(z)+p(z)q(z)\not\equiv 0$. Differential \eqref{e5.1} and
eliminating $e^{^{q(z)}}$, we get
\begin{align*}
&\frac{(f(z)(f(z)-1)\prod_{i=1}^{n}(f(z+c_i)-b_j))'}{f(z)(f(z)-1)
 \prod_{i=1}^{n}(f(z+c_i)-b_j)}\\
&=\frac{p'(z)+p(z)q'(z)}{p(z)}-b\frac{p'(z)+p(z)q'(z)}{p(z)f(z)(f(z)-1)
 \prod_{i=1}^{n}(f(z+c_i)-b_j)}
\end{align*}
Since $f(z)$ or $f(z)-1$ has infinitely many multi-order zeros,
there exists a sufficiently large point $z_0$ such that the multiplicity
of the zero of $f(z)(f(z)-1)$  at $z_0$ is $k$ $(k\geq 2)$, and
$p'(z_0)+p(z_0)q(z_0)\neq 0$, $p(z_0)\neq 0$.
We can easily obtain that the multiplicity of
\[
\frac{(f(z_0)(f(z_0)-1)\prod_{i=1}^{n}(f(z_0+c_i)-b_i))'}
{f(z_0)(f(z_0)-1)\prod_{i=1}^{n}(f(z_0+c_i)-b_i)}=\infty
\]
is 1, and the multiplicity of
$$
\frac{p'(z_0)+p(z)q'(z_0)}{p(z_0)}-b\frac{p'(z_0)+p(z_0)q'(z_0)}
{p(z_0)f(z_0)(f(z_0)-1)\prod_{i=1}^{n}(f(z_0+c_i)-b_i)}=\infty
$$
is $k$ $(l\geq k\geq 2)$. From the above equation, we get a contradiction.
Hence $H(z)$ takes every value $b$ infinitely often.
\end{proof}

\subsection*{Acknowledgements}
The  first author thanks  Prof  Risto Korhonen  for his hospitality 
during the study period in Department of Physics and Mathematics, 
University of Eastern Finland, and for many valuable suggestions to 
the present paper. The authors are also grateful to the anonymous referee
for providing many comments and suggestions for helping us to improve the paper.

This  work was supported by the NNSF of China (No.10771121),
the NSFC Tianyuan Mathematics Youth Fund(No. 11226094),
the NSF of Shandong Province, China (No. ZR2012AQ020 and No. ZR2010AM030)
and the Fund of Doctoral Program Research of University of Jinan
(XBS1211).


\begin{thebibliography}{99}

\bibitem{c1}  Y. M. Chiang, S. J. Feng;
\emph{On the growth of logarithmic differences, difference quotients and 
logarithmic derivatives of meromorphic functions}, 
Trans. Amer. Math. Soc. 361  (2009),  3767-3791.

\bibitem{g1} A. A. Goldberg, I. V. Ostrovskii;
\emph{Distribution of Values of Meromorphic Functions}. Nauka, Mosow, 1970.

\bibitem{h1} R.  G. Halburd, R. Korhonen;
\emph{Nevanlinna theory for the difference operator},  
Ann. Acad. Sci. Fenn. Math. 94 (2006),  463-478.

\bibitem{h2} R. G. Halburd,  R. Korhonen;
\emph{Difference analogue of the lemma on the logarithmic derivative with
applications to difference equatons},  J. Math. Anal. Appl.
314 (2006), 477-487.

\bibitem{h3} R. G. Halburd,  R. Korhonen;
\emph{Finite order solutions and the discrete Painlev\'{e} equations}, Proc.
London Math. Soc. 94 (2007), 443-474.

\bibitem{h4} R. G. Halburd,  R. Korhonen;
\emph{Meromorphic solutions of difference equations, integrability 
and the discrete  Painlev\'{e} equations}. J- Phys. A. 40 (2007), 1-38.

\bibitem{h5} W. Hayman;
\emph{Meromorphic Functions}, Clarendon Press, Oxford, 1964.

\bibitem{h6} J. Heittokangas, R. Korhonen, I. Laine, Rieppo, J. Zhang;
\emph{Value sharing results for shift of meromorphic functions, 
and sufficent conditions for periodicity}. J. Math. Anal.
Appl. 355 (2009), 352-363.

\bibitem{i1} K. Ishizaki,  N. Yanagihara;
\emph{Wiman-Valiron method for difference equations},
 Nagoya Math. J. 175 (2004), 75-102.

\bibitem{y1} C. C. Yang, H. X. Yi;
\emph{Uniqueness of Meromorphic Functions}, Kluwer, Dordrecht, 2003.

\bibitem{y2} H. X. Yi, C. C. Yang;
\emph{Unicity theorems for two meromorphic functions with their first 
derivatives have the same 1 points}. Acta Math. Sinica 34(5) (1991), 675-680.

\bibitem{y3} H. X. Yi;
\emph{Meromorphic functions that share two or three values}, 
Kodai Math. J. 13 (1990), 363-372.

\bibitem{z1} J. Zhang;
\emph{Value distribution and shared sets of differences of meromorphic functions},
 J. Math. Anal. Appl. 367(2) (2010), 401-408.

\end{thebibliography}

\end{document}