- Research
- Open Access
- Published:

# Value distribution of difference and *q*-difference polynomials

*Advances in Difference Equations*
**volume 2013**, Article number: 98 (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 $\alpha (z)\not\equiv 0$ be a small function of *f*, and let

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

**MSC:**30D35, 39A05.

## 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 [1–3]. A function $f(z)$ 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 $S(r,f)$ any quantity satisfying

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

Recently, a number of papers concerning the complex difference products and the differences analogues of Nevanlinna’s theory have been published (see [4–12] 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* $f(z)$ *be a transcendental entire function of finite order*, *and* *c* *be a non*-*zero complex constant*. *Then for* $n\ge 2$, $f{(z)}^{n}f(z+c)$ *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* $f(z)$ *be a transcendental entire function of finite order*, *and* *c* *be a non*-*zero complex constant*. *Then for* $n\ge 2$, $f{(z)}^{n}f(z+c)-p(z)$ *has infinitely many zeros*, *where* $p(z)\not\equiv 0$ *is a polynomial in* *z*.

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

For the sake of simplicity, we denote by $s(P)$ and $m(P)$ the number of the simple zeros and the number of multiple zeros of a polynomial

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 (f)=\sigma $, *and* *c* *be a non*-*zero constant*, *and let*

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

**Remark 1** The result of Theorem 1.1 may be false if $\alpha (z)\equiv 0$, for example, $f(z)=\frac{{e}^{{z}^{2}}}{z}$, it is obvious that ${f}^{2}f(z+1)$ has no zeros. The following example shows that the assumption $\lambda (\frac{1}{f})<\sigma $ in Theorem 1.1 cannot be deleted. In fact, let $f(z)=\frac{1-{e}^{z}}{1+{e}^{z}}$, $c=\pi i$, $\alpha (z)=-1$, and $P(z)={z}^{2}$. Then $\lambda (\frac{1}{f})=\sigma (f)=1$ and $P(f)f(z+c)-\alpha (z)=\frac{2}{1+{e}^{z}}$ has no zeros. Also, let $f(z)=i+{e}^{z}$, $c=\pi i$, $\alpha (z)=1$, and $P(z)=z(z-i+1)(z-i-1)$. Then $P(f)f(z+c)-\alpha (z)=-{e}^{4z}$ 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* $f(z)$ *be a transcendental entire function of zero order*, *and* $\alpha (z)\in S(r,f)$. *Suppose that* *q* *is a non*-*zero complex constant and* *n* *is an integer*. *If* $m(P)>0$, *then* $P(f)f(qz)-\alpha (z)$ *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*

**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*

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

**Lemma 2.3**
*Let*
$f(z)$
*be a transcendental meromorphic solution of*

*where* $A(z,f)$, $B(z,f)$ *are differential polynomials in* *f* *and its derivatives with meromorphic coefficients*, *say* $\{{a}_{\lambda}\mid \lambda \in I\}$, *n* *be a positive integer*. *If the total degree of* $B(z,f)$ *as a polynomial in* *f* *and its derivatives is less than or equal to* *n*, *then*

**Lemma 2.4** [12]

*Let* $f(z)$ *be a non*-*constant meromorphic function of finite order*, $c\in \mathbb{C}$. *Then*

*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* $q\in \mathbb{C}\setminus \{0\}$. *Then*

*on a set of logarithmic density* 1.

**Remark 2** For the similar reason in Theorem 1.1 in [4], we can easily deduce that

also holds on a set of logarithmic density 1.

*Proof*

Using the identity

and let Poisson-Jensen formula with $R=\rho $, we see

where $\{{a}_{n}\}$ and $\{{b}_{m}\}$ are the zeros and poles of *f*, respectively. Integration on the set $E:=\{\phi \in [0,2\pi ]:|\frac{f(r{e}^{i\phi})}{f(qr{e}^{i\phi})}|\ge 1\}$ gives us the proximity function,

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* $q\in \mathbb{C}\setminus \{0\}$. *Then*

*on a set of logarithmic density* 1.

*Proof* Since *f* is an entire function of zero-order, we deduce from Lemma 2.5 that

that is

On the other hand, using Remark 2, we get

that is

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

## 3 Proof of Theorem 1.1

Let $\beta (z)$ be the canonical products of the nonzero poles of $P(f)f(z+c)-\alpha (z)$. Since $\lambda (1/f)<\sigma $ and $\alpha (z)$ is a small function of $f(z)$, we know that $\sigma (\beta )=\lambda (\beta )<\sigma (f)$. Suppose on contrary to the assertion that $P(f)f(z+c)-\alpha (z)$ has finitely many zeros. Then we have

where $Q(z)$ is a polynomial, and $R(z)\not\equiv 0$ is a rational function. Set $H(z)=R(z)/\beta (z)$. Then

and

Differentiating (3.2) and eliminating ${e}^{Q(z)}$, we obtain

Let ${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{t}$ be the distinct zeros of $P(z)$. Then

Substituting this into (3.3), we have

Note that $P(z)$ has at least one multiple zero, we may assume that ${n}_{1}>1$ without loss of generality, and we have

where

Now we distinguish two cases.

*Case* 1. $F(z,f)\equiv 0$. In this case, we obtain from (3.4) that

Since $\alpha (z)\not\equiv 0$ and $H(z)\not\equiv 0$, by integrating, we have

where *k* is a non-zero constant. From (3.2) and (3.5), we have

By Lemma 2.2, we have

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

*Case* 2. $F(z,f)\not\equiv 0$. In this case, we set

Since $f(z)=(f(z)-{\alpha}_{1})+{\alpha}_{1}$ and ${f}^{(k)}={(f-{\alpha}_{1})}^{(k)}$, we know that ${F}^{\ast}(z,f)$ is a differential polynomial of $f(z)-{\alpha}_{1}$ with meromorphic coefficients, and

By Lemma 2.3, we have

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

The lemma of logarithmic derivative implies that

It follows from (3.7) to (3.10) that

Since $(f-{\alpha}_{1}){F}^{\ast}(z,f)=F(z,f)$, we obtain from the definition of $F(z,f)$ that

Thus,

Note that, a zero of $f(z)-{\alpha}_{1}$ which is not a pole of $f(z+c)$ and $H(z)$, is a pole of ${F}^{\ast}(z,f)$ with the multiplicity at most 1, we know from (3.6), (3.1), Lemma 2.4 and $\lambda (1/f)<\sigma $ that

for the positive *ε* sufficiently small. Hence (see the definition of ${F}^{\ast}(z,f)$),

It follows from (3.15) and (3.11) that

Thus, we deduce from (3.16) and (3.13) that

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

## 4 Proof of Theorem 1.2

Denote $F(z)=P(f)f(qz)$. From Lemma 2.6 and the standard Valiron-Mohon’ko theorem, we deduce

Since *f* is a entire function, then by the second main theorem and Lemma 2.5, we have

that is,

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

## References

- 1.
Hayman WK:

*Meromorphic Functions*. Clarendon, Oxford; 1964. - 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. - 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 $f(z+\eta )$ 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

## 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).

## Author information

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors drafted the manuscript, read and approved the final manuscript.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

#### Received

#### Accepted

#### Published

#### DOI

### Keywords

- meromorphic functions
- difference polynomials
- uniqueness