Impact Factor 0.335

Open Access

# Value distribution of difference and q-difference polynomials

DOI: 10.1186/1687-1847-2013-98

Accepted: 25 March 2013

Published: 10 April 2013

## Abstract

In this paper, we investigate the value distribution of difference polynomial and obtain the following result, which improves a recent result of K. Liu and L.Z. Yang: Let f be a transcendental meromorphic function of finite order σ, c be a nonzero constant, and $\alpha \left(z\right)\not\equiv 0$ be a small function of f, and let

$P\left(z\right)={a}_{n}{z}^{n}+{a}_{n-1}{z}^{n-1}+\cdots +{a}_{1}z+{a}_{0}$

be a polynomial with a multiple zero. If $\lambda \left(1/f\right)<\sigma$, then $P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)$ has infinitely many zeros. We also obtain a result concerning the value distribution of q-difference polynomial.

MSC:30D35, 39A05.

### Keywords

meromorphic functions difference polynomials uniqueness

## 1 Introduction and main results

Throughout the paper, we assume that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory as found in [13]. A function $f\left(z\right)$ is called the meromorphic function, if it is analytic in the complex plane except at isolated poles. For any non-constant meromorphic function f, we denote by $S\left(r,f\right)$ any quantity satisfying
$\underset{r\to \mathrm{\infty }}{lim}\frac{S\left(r,f\right)}{T\left(r,f\right)}=0,$

possibly outside of a set of finite linear measure in ${\mathbb{R}}^{+}$. A meromorphic function $a\left(z\right)$ is called a small function of $f\left(z\right)$ provided that $T\left(r,a\right)=S\left(r,f\right)$. As usual, we denote by $\sigma \left(f\right)$ the order of a meromorphic function $f\left(z\right)$, and denote by $\lambda \left(f\right)$ ($\lambda \left(1/f\right)$) the exponent of convergence of the zeros (poles) of $f\left(z\right)$.

Recently, a number of papers concerning the complex difference products and the differences analogues of Nevanlinna’s theory have been published (see [412] for example), and many excellent results have been obtained. In 2007, Laine and Yang [10] investigated the value distribution of difference products of entire functions, and obtained the following result.

Theorem A Let $f\left(z\right)$ be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for $n\ge 2$, $f{\left(z\right)}^{n}f\left(z+c\right)$ assumes every non-zero value $a\in \mathbb{C}$ infinitely often.

Liu and Yang [11] improved Theorem A, and proved the next result.

Theorem B Let $f\left(z\right)$ be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for $n\ge 2$, $f{\left(z\right)}^{n}f\left(z+c\right)-p\left(z\right)$ has infinitely many zeros, where $p\left(z\right)\not\equiv 0$ is a polynomial in z.

The purpose of this paper is to investigate the value distribution of difference polynomial $P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)$ and q-difference polynomial $P\left(f\right)f\left(qz\right)-\alpha \left(z\right)$, where $P\left(z\right)={a}_{n}{z}^{n}+{a}_{n-1}{z}^{n-1}+\cdots +{a}_{1}z+{a}_{0}$ with constant coefficients ${a}_{n}\phantom{\rule{0.25em}{0ex}}\left(\ne 0\right),{a}_{n-1},\dots ,{a}_{0}$, and $\alpha \left(z\right)$ is a mall function of $f\left(z\right)$.

For the sake of simplicity, we denote by $s\left(P\right)$ and $m\left(P\right)$ the number of the simple zeros and the number of multiple zeros of a polynomial
$P\left(z\right)={a}_{n}{z}^{n}+{a}_{n-1}{z}^{n-1}+\cdots +{a}_{1}z+{a}_{0}$

respectively.

We obtain the following result which improves Theorem A and Theorem B.

Theorem 1.1 Let f be a transcendental meromorphic function of finite order $\sigma \left(f\right)=\sigma$, and c be a non-zero constant, and let
$P\left(z\right)={a}_{n}{z}^{n}+{a}_{n-1}{z}^{n-1}+\cdots +{a}_{1}z+{a}_{0}$

be a polynomial with constant coefficients ${a}_{n}\phantom{\rule{0.25em}{0ex}}\left(\ne 0\right),{a}_{n-1},\dots ,{a}_{0}$ and $m\left(P\right)>0$. If $\lambda \left(\frac{1}{f}\right)<\sigma$, then $P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)$ has infinitely many zeros, where $\alpha \left(z\right)\not\equiv 0$ is a small function of f.

Remark 1 The result of Theorem 1.1 may be false if $\alpha \left(z\right)\equiv 0$, for example, $f\left(z\right)=\frac{{e}^{{z}^{2}}}{z}$, it is obvious that ${f}^{2}f\left(z+1\right)$ has no zeros. The following example shows that the assumption $\lambda \left(\frac{1}{f}\right)<\sigma$ in Theorem 1.1 cannot be deleted. In fact, let $f\left(z\right)=\frac{1-{e}^{z}}{1+{e}^{z}}$, $c=\pi i$, $\alpha \left(z\right)=-1$, and $P\left(z\right)={z}^{2}$. Then $\lambda \left(\frac{1}{f}\right)=\sigma \left(f\right)=1$ and $P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)=\frac{2}{1+{e}^{z}}$ has no zeros. Also, let $f\left(z\right)=i+{e}^{z}$, $c=\pi i$, $\alpha \left(z\right)=1$, and $P\left(z\right)=z\left(z-i+1\right)\left(z-i-1\right)$. Then $P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)=-{e}^{4z}$ has no zeros. This shows that the restriction in Theorem 1.1 to the multiple zero case is essential.

Considering the value distribution of q-differences polynomials, we obtain the following result.

Theorem 1.2 Let $f\left(z\right)$ be a transcendental entire function of zero order, and $\alpha \left(z\right)\in S\left(r,f\right)$. Suppose that q is a non-zero complex constant and n is an integer. If $m\left(P\right)>0$, then $P\left(f\right)f\left(qz\right)-\alpha \left(z\right)$ has infinitely many zeros.

## 2 Some lemmas

Lemma 2.1 [6]

Given two distinct complex constants ${\eta }_{1}$, ${\eta }_{2}$, let f be a meromorphic function of finite order σ. Then, for each $\epsilon >0$, we have
$m\left(r,\frac{f\left(z+{\eta }_{1}\right)}{f\left(z+{\eta }_{2}\right)}\right)=O\left({r}^{\sigma -1+\epsilon }\right).$

Lemma 2.2 [6]

Let f be a transcendental meromorphic function of finite order σ, c be a complex number. Then, for each $\epsilon >0$, we have
$T\left(r,f\left(z+c\right)\right)=T\left(r,f\left(z\right)\right)+O\left({r}^{\sigma -1+\epsilon }\right)+O\left(logr\right).$

The following lemma is a revised form of Lemma 2.4.2 in [2].

Lemma 2.3 Let $f\left(z\right)$ be a transcendental meromorphic solution of
${f}^{n}A\left(z,f\right)=B\left(z,f\right),$
where $A\left(z,f\right)$, $B\left(z,f\right)$ are differential polynomials in f and its derivatives with meromorphic coefficients, say $\left\{{a}_{\lambda }\mid \lambda \in I\right\}$, n be a positive integer. If the total degree of $B\left(z,f\right)$ as a polynomial in f and its derivatives is less than or equal to n, then
$m\left(r,A\left(z,f\right)\right)\le \sum _{\lambda \in I}m\left(r,{a}_{\lambda }\right)+S\left(r,f\right).$

Lemma 2.4 [12]

Let $f\left(z\right)$ be a non-constant meromorphic function of finite order, $c\in \mathbb{C}$. Then
$\begin{array}{c}N\left(r,\frac{1}{f\left(z+c\right)}\right)\le N\left(r,\frac{1}{f\left(z\right)}\right)+S\left(r,f\right),\phantom{\rule{2em}{0ex}}N\left(r,f\left(z+c\right)\right)\le N\left(r,f\right)+S\left(r,f\right),\hfill \\ \overline{N}\left(r,\frac{1}{f\left(z+c\right)}\right)\le \overline{N}\left(r,\frac{1}{f\left(z\right)}\right)+S\left(r,f\right),\phantom{\rule{2em}{0ex}}\overline{N}\left(r,f\left(z+c\right)\right)\le \overline{N}\left(r,f\right)+S\left(r,f\right),\hfill \end{array}$

outside of a possible exceptional set E with finite logarithmic measure.

Lemma 2.5 [4]

Let f be a non-constant zero-order meromorphic function, and $q\in \mathbb{C}\setminus \left\{0\right\}$. Then
$m\left(r,\frac{f\left(qz\right)}{f\left(z\right)}\right)=o\left(T\left(r,f\right)\right)$

on a set of logarithmic density 1.

Remark 2 For the similar reason in Theorem 1.1 in [4], we can easily deduce that
$m\left(r,\frac{f\left(z\right)}{f\left(qz\right)}\right)=o\left(T\left(r,f\right)\right)$

also holds on a set of logarithmic density 1.

Proof

Using the identity
$\frac{{\rho }^{2}-{r}^{2}}{{\rho }^{2}-2\rho rcos\left(\phi -\theta \right)+{r}^{2}}=Re\left(\frac{\rho {e}^{i\theta }+z}{\rho {e}^{i\theta }-z}\right),$
and let Poisson-Jensen formula with $R=\rho$, we see
$\begin{array}{rl}log|\frac{f\left(z\right)}{f\left(qz\right)}|=& {\int }_{0}^{2\pi }log|f\left(\rho {e}^{i\theta }\right)|Re\left(\frac{\rho {e}^{i\theta }+z}{\rho {e}^{i\theta }-z}-\frac{\rho {e}^{i\theta }+qz}{\rho {e}^{i\theta }-qz}\right)\frac{d\theta }{2\pi }\\ +\sum _{|{a}_{n}|<\rho }log|\frac{\left(z-{a}_{n}\right)\left({\rho }^{2}-{\overline{a}}_{n}qz\right)}{\left(qz-{a}_{n}\right)\left({\rho }^{2}-{\overline{a}}_{n}z\right)}|\\ -\sum _{|{b}_{m}|<\rho }log|\frac{\left(z-{b}_{m}\right)\left({\rho }^{2}-{\overline{b}}_{m}qz\right)}{\left(qz-{b}_{m}\right)\left({\rho }^{2}-{\overline{b}}_{m}z\right)}|\\ =& {S}_{1}^{\prime }\left(z\right)+{S}_{1}^{\prime }\left(z\right)-{S}_{3}^{\prime }\left(z\right),\end{array}$
where $\left\{{a}_{n}\right\}$ and $\left\{{b}_{m}\right\}$ are the zeros and poles of f, respectively. Integration on the set $E:=\left\{\phi \in \left[0,2\pi \right]:|\frac{f\left(r{e}^{i\phi }\right)}{f\left(qr{e}^{i\phi }\right)}|\ge 1\right\}$ gives us the proximity function,
$\begin{array}{rl}m\left(r,\frac{f\left(z\right)}{f\left(qz\right)}\right)& ={\int }_{E}log|\frac{f\left(z\right)}{f\left(qz\right)}|\frac{d\psi }{2\pi }\\ ={\int }_{E}\left({S}_{1}^{\prime }\left(r{e}^{i\psi }\right)+{S}_{2}^{\prime }\left(r{e}^{i\psi }\right)-{S}_{3}^{\prime }\left(r{e}^{i\psi }\right)\right)\frac{d\psi }{2\pi }\\ \le {\int }_{0}^{2\pi }\left(|{S}_{1}^{\prime }\left(r{e}^{i\psi }\right)|+|{S}_{2}^{\prime }\left(r{e}^{i\psi }\right)|+|{S}_{3}^{\prime }\left(r{e}^{i\psi }\right)|\right)\frac{d\psi }{2\pi }.\end{array}$

Since ${S}_{i}^{\prime }=-{S}_{i}$ ($i=1,2,3$) in [4], we get $|{S}_{i}^{\prime }|=|{S}_{i}|$ ($i=1,2,3$).

Following the similar method in the proof of Theorem 1.1 in [4], we get the result. □

Lemma 2.6 Let f be a non-constant zero-order entire function, and $q\in \mathbb{C}\setminus \left\{0\right\}$. Then
$T\left(r,P\left(f\right)f\left(qz\right)\right)=T\left(r,P\left(f\right)f\left(z\right)\right)+S\left(r,f\right)$

on a set of logarithmic density 1.

Proof Since f is an entire function of zero-order, we deduce from Lemma 2.5 that
$\begin{array}{rl}T\left(r,P\left(f\right)f\left(qz\right)\right)& =m\left(r,P\left(f\right)f\left(qz\right)\right)\\ \le m\left(r,P\left(f\right)f\left(z\right)\right)+m\left(r,\frac{f\left(qz\right)}{f\left(z\right)}\right)\\ \le m\left(r,P\left(f\right)f\left(z\right)\right)+S\left(r,f\right)\\ =T\left(r,P\left(f\right)f\left(z\right)\right)+S\left(r,f\right),\end{array}$
that is
$T\left(r,P\left(f\right)f\left(qz\right)\right)\le T\left(r,P\left(f\right)f\left(z\right)\right)+S\left(r,f\right).$
(2.1)
On the other hand, using Remark 2, we get
$\begin{array}{rl}T\left(r,P\left(f\right)f\left(z\right)\right)& =m\left(r,P\left(f\right)f\left(z\right)\right)\\ \le m\left(r,P\left(f\right)f\left(qz\right)\right)+m\left(r,\frac{f\left(z\right)}{f\left(qz\right)}\right)\\ \le m\left(r,P\left(f\right)f\left(qz\right)\right)+S\left(r,f\right)\\ =T\left(r,P\left(f\right)f\left(qz\right)\right)+S\left(r,f\right),\end{array}$
that is
$T\left(r,P\left(f\right)f\left(z\right)\right)\le T\left(r,P\left(f\right)f\left(qz\right)\right)+S\left(r,f\right).$
(2.2)

The assertion follows from (2.1) and (2.2). □

## 3 Proof of Theorem 1.1

Let $\beta \left(z\right)$ be the canonical products of the nonzero poles of $P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)$. Since $\lambda \left(1/f\right)<\sigma$ and $\alpha \left(z\right)$ is a small function of $f\left(z\right)$, we know that $\sigma \left(\beta \right)=\lambda \left(\beta \right)<\sigma \left(f\right)$. Suppose on contrary to the assertion that $P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)$ has finitely many zeros. Then we have
$P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)=R\left(z\right){e}^{Q\left(z\right)}/\beta \left(z\right),$
where $Q\left(z\right)$ is a polynomial, and $R\left(z\right)\not\equiv 0$ is a rational function. Set $H\left(z\right)=R\left(z\right)/\beta \left(z\right)$. Then
$\sigma \left(H\right)<\sigma \left(f\right)=\sigma ,$
(3.1)
and
$P\left(f\right)f\left(z+c\right)-\alpha \left(z\right)=H\left(z\right){e}^{Q\left(z\right)}.$
(3.2)
Differentiating (3.2) and eliminating ${e}^{Q\left(z\right)}$, we obtain
$\begin{array}{r}{P}^{\prime }\left(f\right){f}^{\prime }\left(z\right)f\left(z+c\right)H\left(z\right)+P\left(f\right){f}^{\prime }\left(z+c\right)H\left(z\right)-P\left(f\right)f\left(z+c\right){H}^{\prime }\left(z\right)-P\left(f\right)f\left(z+c\right){Q}^{\prime }\left(z\right)H\left(z\right)\\ \phantom{\rule{1em}{0ex}}={\alpha }^{\prime }\left(z\right)H\left(z\right)-\alpha \left(z\right){H}^{\prime }\left(z\right)-\alpha \left(z\right){Q}^{\prime }\left(z\right)H\left(z\right).\end{array}$
(3.3)
Let ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{t}$ be the distinct zeros of $P\left(z\right)$. Then
$P\left(f\right)={a}_{n}{\left(f-{\alpha }_{1}\right)}^{{n}_{1}}{\left(f-{\alpha }_{2}\right)}^{{n}_{2}}\cdots {\left(f-{\alpha }_{t}\right)}^{{n}_{t}}.$
Substituting this into (3.3), we have
$\begin{array}{r}{a}_{n}\prod _{j=1}^{t}{\left(f-{\alpha }_{j}\right)}^{{n}_{j}-1}\left\{\left({n}_{1}\prod _{j\ne 1}\left(f-{\alpha }_{j}\right)+{n}_{2}\prod _{j\ne 2}\left(f-{\alpha }_{j}\right)+\cdots +{n}_{t}\prod _{j\ne t}\left(f-{\alpha }_{j}\right)\right)\\ \phantom{\rule{2em}{0ex}}×f\left(z+c\right)H\left(z\right){f}^{\prime }\left(z\right)+{f}^{\prime }\left(z+c\right)H\left(z\right)\\ \phantom{\rule{2em}{0ex}}×\prod _{j=1}^{t}\left(f-{\alpha }_{j}\right)-f\left(z+c\right)\left({H}^{\prime }\left(z\right)+{Q}^{\prime }\left(z\right)H\left(z\right)\right)\prod _{j=1}^{t}\left(f-{\alpha }_{j}\right)\right\}\\ \phantom{\rule{1em}{0ex}}={\alpha }^{\prime }\left(z\right)H\left(z\right)-\alpha \left(z\right){H}^{\prime }\left(z\right)-\alpha \left(z\right){Q}^{\prime }\left(z\right)H\left(z\right).\end{array}$
Note that $P\left(z\right)$ has at least one multiple zero, we may assume that ${n}_{1}>1$ without loss of generality, and we have
${a}_{n}{\left(f-{\alpha }_{1}\right)}^{{n}_{1}-1}F\left(z,f\right)={\alpha }^{\prime }\left(z\right)H\left(z\right)-\alpha \left(z\right){H}^{\prime }\left(z\right)-\alpha \left(z\right){Q}^{\prime }\left(z\right)H\left(z\right),$
(3.4)
where
$\begin{array}{rl}F\left(z,f\right)=& \prod _{j=2}^{t}{\left(f-{\alpha }_{j}\right)}^{{n}_{j}-1}\left\{\left({n}_{1}\prod _{j\ne 1}\left(f-{\alpha }_{j}\right)+{n}_{2}\prod _{j\ne 2}\left(f-{\alpha }_{j}\right)+\cdots +{n}_{t}\prod _{j\ne t}\left(f-{\alpha }_{j}\right)\right)\\ ×f\left(z+c\right)H\left(z\right){f}^{\prime }\left(z\right)+{f}^{\prime }\left(z+c\right)H\left(z\right)\prod _{j=1}^{t}\left(f-{\alpha }_{j}\right)\\ -f\left(z+c\right)\left({H}^{\prime }\left(z\right)+{Q}^{\prime }\left(z\right)H\left(z\right)\right)\prod _{j=1}^{t}\left(f-{\alpha }_{j}\right)\right\}.\end{array}$

Now we distinguish two cases.

Case 1. $F\left(z,f\right)\equiv 0$. In this case, we obtain from (3.4) that
${\alpha }^{\prime }\left(z\right)H\left(z\right)-\alpha \left(z\right){H}^{\prime }\left(z\right)-\alpha \left(z\right){Q}^{\prime }\left(z\right)H\left(z\right)\equiv 0.$
Since $\alpha \left(z\right)\not\equiv 0$ and $H\left(z\right)\not\equiv 0$, by integrating, we have
$\frac{\alpha \left(z\right)}{H\left(z\right)}=k{e}^{Q\left(z\right)},$
(3.5)
where k is a non-zero constant. From (3.2) and (3.5), we have
$P\left(f\right)f\left(z+c\right)=\left(\frac{1}{k}+1\right)\alpha \left(z\right).$
By Lemma 2.2, we have
$\begin{array}{rl}nT\left(r,f\left(z\right)\right)& =T\left(r,P\left(f\right)\right)+O\left(1\right)\\ \le T\left(r,f\left(z+c\right)\right)+T\left(r,\alpha \left(z\right)\right)+O\left(1\right)\\ =T\left(r,f\left(z\right)\right)+O\left({r}^{\sigma -1+\epsilon }\right)+S\left(r,f\right).\end{array}$

Since $n\ge {n}_{1}\ge 2$, and $f\left(z\right)$ is a transcendental, this is impossible.

Case 2. $F\left(z,f\right)\not\equiv 0$. In this case, we set
$\begin{array}{rl}{F}^{\ast }\left(z,f\right)=& \frac{F\left(z,f\right)}{f-{\alpha }_{1}}\\ =& \prod _{j=2}^{t}{\left(f-{\alpha }_{j}\right)}^{{n}_{j}-1}\left\{\left({n}_{1}\prod _{j\ne 1}\left(f-{\alpha }_{j}\right)+{n}_{2}\prod _{j\ne 2}\left(f-{\alpha }_{j}\right)+\cdots +{n}_{t}\prod _{j\ne t}\left(f-{\alpha }_{j}\right)\right)\\ ×\frac{f\left(z+c\right)}{f\left(z\right)}f\left(z\right)H\left(z\right)\frac{{f}^{\prime }\left(z\right)}{f-{\alpha }_{1}}+\frac{{f}^{\prime }\left(z+c\right)}{f\left(z+c\right)}\frac{f\left(z+c\right)}{f\left(z\right)}f\left(z\right)H\left(z\right)\prod _{j=2}^{t}\left(f-{\alpha }_{j}\right)\\ -\frac{f\left(z+c\right)}{f\left(z\right)}f\left(z\right)\left({H}^{\prime }\left(z\right)+{Q}^{\prime }\left(z\right)H\left(z\right)\right)\prod _{j=2}^{t}\left(f-{\alpha }_{j}\right)\right\}.\end{array}$
Since $f\left(z\right)=\left(f\left(z\right)-{\alpha }_{1}\right)+{\alpha }_{1}$ and ${f}^{\left(k\right)}={\left(f-{\alpha }_{1}\right)}^{\left(k\right)}$, we know that ${F}^{\ast }\left(z,f\right)$ is a differential polynomial of $f\left(z\right)-{\alpha }_{1}$ with meromorphic coefficients, and
${a}_{n}{\left(f-{\alpha }_{1}\right)}^{{n}_{1}}{F}^{\ast }\left(z,f\right)={\alpha }^{\prime }\left(z\right)H\left(z\right)-\alpha \left(z\right){H}^{\prime }\left(z\right)-\alpha \left(z\right){Q}^{\prime }\left(z\right)H\left(z\right).$
(3.6)
By Lemma 2.3, we have
$\begin{array}{rl}m\left(r,{\left(f-{\alpha }_{1}\right)}^{k}{F}^{\ast }\left(z,f\right)\right)\le & 3m\left(r,\frac{f\left(z+c\right)}{f\left(z\right)}\right)+m\left(r,\frac{{f}^{\prime }\left(z+c\right)}{f\left(z+c\right)}\right)+m\left(r,\frac{{f}^{\prime }\left(z\right)}{f-{\alpha }_{1}}\right)\\ +5T\left(r,H\right)+S\left(r,f\right)\end{array}$
(3.7)

for $k=0$ and $k=1$.

Now for any given ε ($0<\epsilon <1$), we obtain from Lemma 2.1, Lemma 2.2 and (3.1) that
$m\left(r,\frac{f\left(z+c\right)}{f\left(z\right)}\right)=O\left({r}^{\sigma -\epsilon }\right),\phantom{\rule{2em}{0ex}}T\left(r,H\right)=O\left({r}^{\sigma -\epsilon }\right),$
(3.8)
$m\left(r,\frac{{f}^{\prime }\left(z+c\right)}{f\left(z+c\right)}\right)=O\left({r}^{\sigma -\epsilon }\right)+S\left(r,f\right).$
(3.9)
The lemma of logarithmic derivative implies that
$m\left(r,\frac{{f}^{\prime }\left(z\right)}{f-{\alpha }_{1}}\right)=S\left(r,f\right).$
(3.10)
It follows from (3.7) to (3.10) that
$m\left(r,{F}^{\ast }\left(z,f\right)\right)=O\left({r}^{\sigma -\epsilon }\right)+S\left(r,f\right),$
(3.11)
$m\left(r,\left(f-{\alpha }_{1}\right){F}^{\ast }\left(z,f\right)\right)=O\left({r}^{\sigma -\epsilon }\right)+S\left(r,f\right).$
(3.12)
Since $\left(f-{\alpha }_{1}\right){F}^{\ast }\left(z,f\right)=F\left(z,f\right)$, we obtain from the definition of $F\left(z,f\right)$ that
$N\left(r,F\left(z,f\right)\right)=O\left(N\left(r,H\left(z\right)\right)+N\left(r,f\right)\right)=O\left({r}^{\sigma -\epsilon }\right)+S\left(r,f\right).$
Thus,
$T\left(r,\left(f-{\alpha }_{1}\right){F}^{\ast }\left(z,f\right)\right)=O\left({r}^{\sigma -\epsilon }\right)+S\left(r,f\right).$
(3.13)
Note that, a zero of $f\left(z\right)-{\alpha }_{1}$ which is not a pole of $f\left(z+c\right)$ and $H\left(z\right)$, is a pole of ${F}^{\ast }\left(z,f\right)$ with the multiplicity at most 1, we know from (3.6), (3.1), Lemma 2.4 and $\lambda \left(1/f\right)<\sigma$ that
$\begin{array}{rl}\left({n}_{1}-1\right)N\left(r,\frac{1}{f\left(z\right)-{\alpha }_{1}}\right)\le & N\left(r,\frac{1}{{\alpha }^{\prime }\left(z\right)H\left(z\right)-\alpha \left(z\right){H}^{\prime }\left(z\right)-\alpha \left(z\right){Q}^{\prime }\left(z\right)H\left(z\right)}\right)\\ +O\left(N\left(r,f\left(z+c\right)\right)\right)+O\left(N\left(r,H\right)\right)\\ =& O\left({r}^{\sigma -\epsilon }\right)\end{array}$
(3.14)
for the positive ε sufficiently small. Hence (see the definition of ${F}^{\ast }\left(z,f\right)$),
$\begin{array}{rl}N\left(r,{F}^{\ast }\left(z,f\right)\right)& =O\left(N\left(r,\frac{1}{f-{\alpha }_{1}}\right)+N\left(r,f\right)+N\left(r,H\right)\right)\\ =O\left({r}^{\sigma -\epsilon }\right)+S\left(r,f\right).\end{array}$
(3.15)
It follows from (3.15) and (3.11) that
$T\left(r,{F}^{\ast }\left(z,f\right)\right)=O\left({r}^{\sigma -\epsilon }\right)+S\left(r,f\right).$
(3.16)
Thus, we deduce from (3.16) and (3.13) that
$\begin{array}{rl}T\left(r,f\left(z\right)\right)& =T\left(r,f\left(z\right)-{\alpha }_{1}\right)+O\left(1\right)=T\left(r,\frac{\left(f-{\alpha }_{1}\right){F}^{\ast }\left(z,f\right)}{{F}^{\ast }\left(z,f\right)}\right)\\ =O\left({r}^{\sigma -\epsilon }\right)+S\left(r,f\right).\end{array}$

This contradicts that f is of order σ. Theorem 1.1 is proved.

## 4 Proof of Theorem 1.2

Denote $F\left(z\right)=P\left(f\right)f\left(qz\right)$. From Lemma 2.6 and the standard Valiron-Mohon’ko theorem, we deduce
$\begin{array}{rl}T\left(r,F\left(z\right)\right)& =T\left(r,P\left(f\right)f\left(z\right)\right)+S\left(r,f\right)\\ =\left(n+1\right)T\left(r,f\left(z\right)\right)+S\left(r,f\right).\end{array}$
Since f is a entire function, then by the second main theorem and Lemma 2.5, we have
$\begin{array}{rcl}T\left(r,F\left(z\right)\right)& \le & \overline{N}\left(r,F\left(z\right)\right)+\overline{N}\left(r,\frac{1}{F\left(z\right)}\right)+\overline{N}\left(r,\frac{1}{F\left(z\right)-\alpha \left(z\right)}\right)+S\left(r,f\right)\\ \le & \overline{N}\left(r,\frac{1}{P\left(f\right)}\right)+\overline{N}\left(r,\frac{1}{f\left(qz\right)}\right)+\overline{N}\left(r,\frac{1}{F\left(z\right)-\alpha \left(z\right)}\right)+S\left(r,f\right)\\ \le & \left(s\left(P\right)+m\left(P\right)\right)T\left(r,f\left(z\right)\right)+T\left(r,f\left(qz\right)\right)\\ +\overline{N}\left(r,\frac{1}{F\left(z\right)-\alpha \left(z\right)}\right)+S\left(r,f\right)\\ \le & \left(s\left(P\right)+m\left(P\right)\right)T\left(r,f\left(z\right)\right)+m\left(r,\frac{f\left(qz\right)}{f\left(z\right)}\right)+m\left(r,f\left(z\right)\right)\\ +\overline{N}\left(r,\frac{1}{F\left(z\right)-\alpha \left(z\right)}\right)+S\left(r,f\right)\\ \le & \left(s\left(P\right)+m\left(P\right)+1\right)T\left(r,f\left(z\right)\right)+\overline{N}\left(r,\frac{1}{F\left(z\right)-\alpha \left(z\right)}\right)+S\left(r,f\right),\end{array}$
that is,
$\overline{N}\left(r,\frac{1}{F\left(z\right)-\alpha \left(z\right)}\right)\ge \left(n-s\left(P\right)-m\left(P\right)\right)T\left(r,f\left(z\right)\right)+S\left(r,f\left(z\right)\right).$

Since f is a transcendental entire function with $m\left(P\right)>0$, we deduce that $P\left(f\right)f\left(qz\right)-\alpha \left(z\right)$ has infinitely many zeros.

## Declarations

### Acknowledgements

This work was supported by the NSF of Shandong Province, P.R. China (No. ZR2010AM030) and the NNSF of China (No. 11171013 and No. 11041005).

## Authors’ Affiliations

(1)
School of Mathematics, Shandong University

## References

1. Hayman WK: Meromorphic Functions. Clarendon, Oxford; 1964.Google Scholar
2. Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.
3. Yi HX, Yang CC: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.Google Scholar
4. Barnett DC, Halburd RG, Korhonen RJ, Morgan W: Nevanlinna theory for the q -difference operator and meromorphic solutions of q -difference equations. Proc. R. Soc. Edinb. A 2007, 137: 457–474.
5. Bergweiler W, Langley JK: Zeros of difference of meromorphic functions. Math. Proc. Camb. Philos. Soc. 2007, 142: 133–147. 10.1017/S0305004106009777
6. Chiang YM, Feng SJ:On the Nevanlinna characteristic $f\left(z+\eta \right)$ and difference equations in the complex plane. Ramanujan J. 2008, 16: 105–129. 10.1007/s11139-007-9101-1
7. Chiang YM, Feng SJ: On the growth of logarithmic differences, difference quotients and logarithmic derivatices of meromorphic functions. Trans. Am. Math. Soc. 2009, 361(7):3767–3791. 10.1090/S0002-9947-09-04663-7
8. Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 2006, 31: 463–478.
9. Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314: 477–487. 10.1016/j.jmaa.2005.04.010
10. Laine I, Yang CC: Value distribution of difference polynomials. Proc. Jpn. Acad., Ser. A, Math. Sci. 2007, 83: 148–151. 10.3792/pjaa.83.148
11. Liu K, Yang LZ: Value distribution of the difference operator. Arch. Math. 2009, 92: 270–278. 10.1007/s00013-009-2895-x
12. Qi XG, Yang LZ, Liu K: Uniqueness and periodicity of meromorphic functions concerning difference operator. Comput. Math. Appl. 2010, 60(6):1739–1746. 10.1016/j.camwa.2010.07.004