- Research
- Open Access

# Meromorphic functions sharing small functions with their linear difference polynomials

- Sheng Li
^{1}and - BaoQin Chen
^{1}Email author

**2013**:58

https://doi.org/10.1186/1687-1847-2013-58

© Li and Chen; licensee Springer 2013

**Received:**30 November 2012**Accepted:**24 February 2013**Published:**18 March 2013

## Abstract

In this paper, we prove some results on the uniqueness of meromorphic functions sharing small functions CM with their linear difference polynomials. Examples are provided to show the existence of meromorphic functions satisfying the conditions of our results.

**MSC:**30D35, 39A10, 39B32.

## Keywords

- uniqueness
- meromorphic functions
- difference polynomials

## 1 Introduction and main results

Denote by $S(r,f)$ any quantity satisfying $S(r,f)=o(T(r,f))$ as $r\to \mathrm{\infty}$, possibly outside of a set of *r* with a finite logarithmic measure. A meromorphic function $a(z)$ is said to be a small function of $f(z)$ if $T(r,a)=S(r,f)$. In what follows, we use $S(f)$ to denote the set of all small functions of $f(z)$.

For two meromorphic functions $f(z)$ and $g(z)$, and $a\in S(f)\cup S(g)\cup \{\mathrm{\infty}\}$, we say that $f(z)$ and $g(z)$ share *a* CM when $f(z)-a$ and $g(z)-a$ have the same zeros counting multiplicity.

For a nonzero complex constant $c\in \mathbb{C}$, $f(z+c)$ is called a shift of $f(z)$. And a difference monomial of type ${\prod}_{i=1}^{m}{f}^{{n}_{i}}(z+{c}_{i})$ is called a difference product of $f(z)$, where ${c}_{1},\dots ,{c}_{m}\in \mathbb{C}$ and ${n}_{1},\dots ,{n}_{m}\in \mathbb{N}$.

where ${c}_{1},\dots ,{c}_{n}\in \mathbb{C}$, ${a}_{1}(z),\dots ,{a}_{n}(z)\in S(f)$.

In particular, ${\mathrm{\Delta}}_{c}^{n}f(z)={\mathrm{\Delta}}^{n}f(z)$ for the case $c=1$. We point out that a difference operator is just a special linear difference polynomial of $f(z)$ such that the sum of its coefficients equals 0.

The subject on the uniqueness of the entire function $f(z)$ sharing values with its derivative ${f}^{\prime}(z)$ was initiated by Rubel and Yang [4]. For a nonconstant entire function $f(z)$, they proved that $f(z)\equiv {f}^{\prime}(z)$ provided that $f(z)$ and ${f}^{\prime}(z)$ share two distinct finite values CM.

Recently, a number of papers have focused on the Nevanlinna theory with respect to difference operators; see, *e.g.*, the papers [5, 6] by Chiang and Feng and [7, 8] by Halburd and Korhonen. Then, many authors started to investigate the uniqueness of meromorphic functions sharing values or small functions with their shifts (see, *e.g.*, [9–14]) or difference operators (see, *e.g.*, [9, 12]). The following Theorem A is indeed a corollary of Theorem 2.1 in [10] and Theorem 2 in [11].

*Let* $f(z)$ *be a meromorphic function of finite order*, *let* $c\in \mathbb{C}$, *and let* ${a}_{1},{a}_{2}\in S(f)$ *be two distinct periodic functions with period* *c*. *If* $f(z)$ *and* $f(z+c)$ *share* ${a}_{1}$, ${a}_{2}$, ∞ *CM*, *then* $f(z)=f(z+c)$ *for all* $z\in \mathbb{C}$.

Theorem B below is Theorem 1.2 in [9], while Theorem C is Theorem 1.1 in [12].

**Theorem B** ([9])

*Let* $f(z)$ *be a transcendental meromorphic function such that its order of growth* $\rho (f)$ *is not an integer or infinite*, *and let* $c\in \mathbb{C}$ *be a constant such that* $f(z+c)\not\equiv f(z)$. *If* ${\mathrm{\Delta}}_{c}f(z)$ *and* $f(z)$ *share three distinct values* *a*, *b*, ∞ *CM*, *then* $f(z+c)=2f(z)$.

**Theorem C** ([12])

*Let*$f(z)$

*be a nonconstant entire function of finite order*, $c\in \mathbb{C}$,

*and*

*n*

*be a positive integer*.

*Suppose that*$f(z)$

*and*${\mathrm{\Delta}}_{c}^{n}f(z)$

*share two distinct finite values*

*a*,

*b*

*CM and one of the following cases is satisfied*:

- (i)
$ab=0$;

- (ii)
$ab\ne 0$

*and*$\rho (f)\notin \mathbb{N}$.

*Then* $f(z)\equiv {\mathrm{\Delta}}_{c}^{n}f(z)$.

**Remark 1** The methods in [9] and [12] are quite different. Due to a result of Ozawa [15] (he proved that for any given $\rho \in [1,\mathrm{\infty})$, there exists a periodic entire function of order *ρ*), Chen and Yi [9] and Li and Gao [12] gave some examples to show the existence of functions satisfying the conditions of Theorem B and Theorem C respectively.

Considering Theorems A-C, due to some ideas of [9] and [12], we obtain the following result with a quite simple proof.

**Theorem 1.1**

*Let*$f(z)$

*be a meromorphic function of hyper order*${\rho}_{2}(f)<1$,

*let*$L(z,f)$

*be a difference polynomial of*$f(z)$,

*and let*$a,b\in S(f)$

*be two distinct meromorphic functions*.

*Suppose that*$f(z)$

*and*$L(z,f)$

*share*

*a*,

*b*, ∞

*CM and one of the following cases holds*:

- (i)
$L(z,a)-a=L(z,b)-b\equiv 0$;

- (ii)
$L(z,a)-a\equiv 0$

*or*$L(z,b)-b\equiv 0$,*and*$N(r,f)<\lambda T(r,f)$*for some*$\lambda \in (0,1)$; - (iii)
$\rho (f)\notin \mathbb{N}\cup \{\mathrm{\infty}\}$.

*Then* $f(z)\equiv L(z,f)$.

**Example 1**We give two examples for Theorem 1.1.

- (1)
For the cases (i) and (ii): Let $f(z)={e}^{zlog3}$ and $L(z,f)=\mathrm{\Delta}f(z)-f(z)=f(z+1)-2f(z)$. Then $L(z,f)=f(z)$, and hence for $a=0$ and any given $b\in S(f)$, $f(z)$ and $L(z,f)$ share

*a*,*b*, ∞ CM. - (2)
For the case (iii): Let $f(z)=g(z){e}^{zlog3}$ and $L(z,f)=\mathrm{\Delta}f(z)-f(z)=f(z+1)-2f(z)$, where $g(z)$ is a periodic entire function with period 1 such that $\rho (g)\in (1,\mathrm{\infty})\setminus \mathbb{N}$. Then $L(z,f)=f(z)$, and hence for any given $a,b\in S(f)$, $f(z)$ and $L(z,f)$ share

*a*,*b*, ∞ CM.

For one CM shared value case, Li and Gao [12] proved the following results.

**Theorem D** ([12])

*Let* $f(z)$ *be a nonconstant entire function of finite order* $\rho (f)$, $\eta \in \mathbb{C}$. *If* $f(z)$ *and* $f(z+\eta )$ *share one finite value* *a* *CM*, *and for a finite value* $b\ne a$, $f(z)-b$ *and* $f(z+\eta )-b$ *have* $max\{1,[\rho (f)]-1\}$ *distinct common zeros of multiplicity* ≥2, *then* $f(z)\equiv f(z+\eta )$.

**Theorem E** ([12])

*Let* $f(z)$ *be a nonconstant entire function of finite order* $\rho (f)$, $\eta \in \mathbb{C}$, *and* *n* *be a positive integer*. *If* $f(z)$ *and* ${\mathrm{\Delta}}_{\eta}^{n}f(z)$ *share one finite value* *a* *CM*, *and for a finite value* $b\ne a$, $f(z)-b$ *and* ${\mathrm{\Delta}}_{\eta}^{n}f(z)-b$ *have* $max\{1,[\rho (f)]\}$ *distinct common zeros of multiplicity* ≥2, *then* $f(z)\equiv {\mathrm{\Delta}}_{\eta}^{n}f(z)$.

To generalize Theorems D and E, we prove Theorem 1.2 below.

**Theorem 1.2** *Let* $f(z)$ *be a meromorphic function of finite order* $\rho (f)$, *let* $L(z,f)$ *be a difference polynomial of* $f(z)$, *and let* $a,b\in S(f)$ *be two distinct meromorphic functions*. *Suppose that* $f(z)$ *and* $L(z,f)$ *share* *a*, ∞ *CM and* $f(z)-b$ *and* $L(z,f)-b$ *have* $m=max\{1,[\rho (f)]\}$ *distinct common zeros of multiplicity* ≥2, *denoted by* ${z}_{1},{z}_{2},\dots ,{z}_{m}$, *such that* $a({z}_{i})\ne b({z}_{i})$. *Then* $f(z)\equiv L(z,f)$.

**Example 2** Let $f(z)={g}^{2}(z){e}^{zlog3}$ and $L(z,f)=\mathrm{\Delta}f(z)-f(z)=f(z+1)-2f(z)$, where $g(z)$ is a periodic entire function with period 1 such that $\rho (g)\in (1,\mathrm{\infty})\setminus \mathbb{N}$. Then $L(z,f)=f(z)$, and hence for any given $a\in S(f)$ and $b=0$, $f(z)$ and $L(z,f)$ share *a*, ∞ CM.

**Remark 2** Chen and Yi [9] (resp. Li and Gao [12]) conjectured that the condition on the order of growth of $f(z)$ in Theorem B (resp. Theorem C) could be omitted. The same conjecture should be made for Theorems 1.1 and 1.2.

## 2 Lemmas

**Lemma 2.1** ([16])

*Let*$f(z)$

*be a meromorphic function of hyper order*${\rho}_{2}(f)=\varsigma <1$, $c\in \mathbb{C}$,

*and*$\epsilon >0$.

*Then*

*possibly outside of a set of* *r* *with a finite logarithmic measure*.

The following lemma is a Clunie-type lemma [17] for the difference-differential polynomials of a meromorphic function *f*, which is a finite sum of products of *f*, derivatives of *f*, and of their shifts, with all the coefficients being small functions of *f*. It can be proved by applying Lemma 2.1 with a similar reasoning as in [18] and stated as follows.

**Lemma 2.2** ([18])

*Let*$f(z)$

*be a meromorphic function of hyper order*${\rho}_{2}(f)<1$

*and*$P(z,f)$, $Q(z,f)$

*be two difference*-

*differential polynomials of*

*f*.

*If*

*holds and if the total degree of* $Q(z,f)$ *in* *f* *and its derivatives and their shifts is* ≤*n*, *then* $m(r,P(z,f))=S(r,f)$.

## 3 Proof of Theorem 1.1

*a*,

*b*, ∞ CM, we have

where $p=p(z)$, $q=q(z)$ are entire functions such that $max\{\rho ({e}^{p}),\rho ({e}^{q})\}\le {\rho}_{2}(f)$.

Since $a-b\not\equiv 0$, we get ${e}^{p}\equiv 1$ and hence finish our proof from (3.1).

Next, we assume that ${e}^{p}\not\equiv {e}^{q}$ and complete our proof in three steps.

which gives $\rho (f)\le max\{degp(z),degq(z)\}$, a contradiction to (3.5).

Step 3. We prove the case (ii): $L(z,a)-a\equiv 0$ or $L(z,b)-b\equiv 0$, and $N(r,f)<\lambda T(r,f)$ for some $\lambda \in (0,1)$. Without loss of generality, we assume that $L(z,a)-a\equiv 0$ and hence (3.4) still holds.

where ${A}_{1}={p}^{\prime}-{q}^{\prime}$, ${A}_{2}=b-a$, ${A}_{3}={b}^{\prime}-{a}^{\prime}+a{p}^{\prime}+b{q}^{\prime}$, ${A}_{4}={a}^{\prime}-{b}_{a}^{\mathrm{\prime}}{p}^{\prime}+b{q}^{\prime}$, ${A}_{5}={q}^{\prime}{b}^{2}-{p}^{\prime}{a}^{2}$.

*f*with degree in

*f*, its derivatives and their shifts being ≤1. Then from Lemma 2.2 and its remark, we have $m(r,L)=S(r,f)$. Considering this, with (3.1) and (3.4), we obtain

and hence $T(r,f)=N(r,f)+m(r,f)=N(r,f)+S(r,f)$, which contradicts the condition $N(r,f)<\lambda T(r,f)$ for some $\lambda \in (0,1)$.

## 4 Proof of Theorem 1.2

*a*CM, we have

where *p* is a polynomial such that $degp(z)\le max\{1,[\rho (f)]\}=m$.

*m*zeros. This means that ${p}^{\prime}(z)\equiv 0$. Therefore,

holds for some nonconstant *c*. For the point ${z}_{1}$ such that $L({z}_{1},f)=f({z}_{1})=b({z}_{1})\ne a({z}_{1})$, we get from (4.5) that $c=1$ and hence prove that $f(z)\equiv L(z,f)$.

## Declarations

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11226091, No. 11171013).

## Authors’ Affiliations

## References

- Hayman WK:
*Meromorphic Functions*. Clarendon Press, Oxford; 1964.Google Scholar - Laine I:
*Nevanlinna Theory and Complex Differential Equations*. de Gruyter, Berlin; 1993.View ArticleGoogle Scholar - Yang CC, Yi HX:
*Uniqueness Theory of Meromorphic Functions*. Kluwer Academic, Dordrecht; 2003.View ArticleGoogle Scholar - Rubel LA, Yang CC Lecture Notes in Math. 599. In
*Values Shared by an Entire Function and Its Derivative*. Springer, Berlin; 1977:101-103.Google Scholar - 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/s11139-007-9101-1MathSciNetView ArticleGoogle Scholar - Chiang YM, Feng SJ: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions.
*Trans. Am. Math. Soc.*2009, 361: 3767-3791. 10.1090/S0002-9947-09-04663-7MathSciNetView ArticleGoogle Scholar - 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.2005.04.010MathSciNetView ArticleGoogle Scholar - Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator.
*Ann. Acad. Sci. Fenn., Math.*2006, 31: 463-478.MathSciNetGoogle Scholar - Chen ZX, Yi HX: On sharing values of meromorphic functions and their differences.
*Results Math.*2013, 63: 557-565. 10.1007/s00025-011-0217-7MathSciNetView ArticleGoogle Scholar - Heittokangas J, Korhonen R, Laine I, Rieppo J: Uniqueness of meromorphic functions sharing values with their shifts.
*Complex Var. Elliptic Equ.*2011, 56: 81-92. 10.1080/17476930903394770MathSciNetView ArticleGoogle Scholar - Heittokangas J, Korhonen R, Laine I, Rieppo J, Zhang J: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity.
*J. Math. Anal. Appl.*2009, 355: 352-363. 10.1016/j.jmaa.2009.01.053MathSciNetView ArticleGoogle Scholar - Li S, Gao ZS: Entire functions sharing one or two finite values CM with their shifts or difference operators.
*Arch. Math.*2011, 97: 475-483. 10.1007/s00013-011-0324-4View ArticleGoogle Scholar - Qi XG: Value distribution and uniqueness of difference polynomials and entire solutions of difference equations.
*Ann. Pol. Math.*2011, 102: 129-142. 10.4064/ap102-2-3View ArticleGoogle Scholar - Qi XG, Liu K: Uniqueness and value distribution of differences of entire functions.
*J. Math. Anal. Appl.*2011, 379: 180-187. 10.1016/j.jmaa.2010.12.031MathSciNetView ArticleGoogle Scholar - Ozawa M: On the existence of prime periodic entire functions.
*Kodai Math. Semin. Rep.*1978, 29: 308-321. 10.2996/kmj/1138833654View ArticleGoogle Scholar - Halburd, RG, Korhonen, RJ, Tohge, K: Holomorphic curves with shift-invariant hyperplane preimages. arXiv:0903.3236Google Scholar
- Clunie J: On integral and meromorphic functions.
*J. Lond. Math. Soc.*1962, 37: 17-27. 10.1112/jlms/s1-37.1.17MathSciNetView ArticleGoogle Scholar - Yang CC, Laine I: On analogies between nonlinear difference and differential equations.
*Proc. Jpn. Acad., Ser. A, Math. Sci.*2010, 86: 10-14. 10.3792/pjaa.86.10MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.