 Research
 Open Access
 Published:
Meromorphic functions sharing small functions with their linear difference polynomials
Advances in Difference Equations volume 2013, Article number: 58 (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.
1 Introduction and main results
In the following, we use the standard notations of Nevanlinna theory of meromorphic functions (see [1–3]). For any given nonconstant meromorphic function $f(z)$, we recall the hyper order of $f(z)$ defined as follows (see [3]):
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 ${n}_{1},\dots ,{n}_{m}\in \mathbb{N}$.
A difference polynomial of $f(z)$ is a finite sum of difference products of $f(z)$, with all coefficients being small functions of $f(z)$. In the following, we mainly consider a linear difference polynomial of $f(z)$ of the form
where ${c}_{1},\dots ,{c}_{n}\in \mathbb{C}$, ${a}_{1}(z),\dots ,{a}_{n}(z)\in S(f)$.
It is well known that the difference operators of $f(z)$ are defined as follows:
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 AC, 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 Clunietype lemma [17] for the differencedifferential 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 differencedifferential 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
Since $f(z)$ and $L:=L(z,f)$ share the value a, b, ∞ CM, we have
and
where $p=p(z)$, $q=q(z)$ are entire functions such that $max\{\rho ({e}^{p}),\rho ({e}^{q})\}\le {\rho}_{2}(f)$.
It follows from (3.1) and (3.2) that
If ${e}^{p}\equiv {e}^{q}$, then from (3.3) we obtain
Since $ab\not\equiv 0$, 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.
Step 1. We prove the case (i): $L(z,a)a=L(z,b)b\equiv 0$. From (3.1), we get from Lemma 2.1 that
Similarly, we have $T(r,{e}^{q})=S(r,f)$. Now, we can deduce a contradiction from (3.3) that
Step 2. We prove the case (iii): $\rho (f)\notin \mathbb{N}\cup \{\mathrm{\infty}\}$. In this case, $p(z)$, $q(z)$ are polynomials, and we have
From (3.3), we obtain
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.
Differentiating (3.1) and (3.2), we get
and
Combining two equations above, we get
where ${A}_{1}={p}^{\prime}{q}^{\prime}$, ${A}_{2}=ba$, ${A}_{3}={b}^{\prime}{a}^{\prime}+a{p}^{\prime}+b{q}^{\prime}$, ${A}_{4}={a}^{\prime}{b}_{a}^{\mathrm{\prime}}{p}^{\prime}+b{q}^{\prime}$, ${A}_{5}={q}^{\prime}{b}^{2}{p}^{\prime}{a}^{2}$.
Notice that the righthand side of (3.6) is a differencedifferential polynomial of 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
Since $f(z)$ and $L(z,f)$ share a CM, we have
where p is a polynomial such that $degp(z)\le max\{1,[\rho (f)]\}=m$.
It follows from (4.1) that
For each point ${z}_{i}$, $1\le i\le m$, satisfying the assumption in Theorem 1.2, we get
and
from (4.1) and (4.3), we see that ${e}^{p({z}_{i})}=1$. Then we can obtain from (4.2) and (4.4) that ${p}^{\prime}({z}_{i})=0$. By assumption, ${p}^{\prime}(z)$ has at least 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 $f(z)\equiv L(z,f)$.
References
 1.
Hayman WK: Meromorphic Functions. Clarendon Press, Oxford; 1964.
 2.
Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.
 3.
Yang CC, Yi HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.
 4.
Rubel LA, Yang CC Lecture Notes in Math. 599. In Values Shared by an Entire Function and Its Derivative. Springer, Berlin; 1977:101103.
 5.
Chiang YM, Feng SJ:On the Nevanlinna characteristic $f(z+\eta )$ and difference equations in the complex plane. Ramanujan J. 2008, 16: 105129. 10.1007/s1113900791011
 6.
Chiang YM, Feng SJ: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Am. Math. Soc. 2009, 361: 37673791. 10.1090/S0002994709046637
 7.
Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314: 477487. 10.1016/j.jmaa.2005.04.010
 8.
Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463478.
 9.
Chen ZX, Yi HX: On sharing values of meromorphic functions and their differences. Results Math. 2013, 63: 557565. 10.1007/s0002501102177
 10.
Heittokangas J, Korhonen R, Laine I, Rieppo J: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. 2011, 56: 8192. 10.1080/17476930903394770
 11.
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: 352363. 10.1016/j.jmaa.2009.01.053
 12.
Li S, Gao ZS: Entire functions sharing one or two finite values CM with their shifts or difference operators. Arch. Math. 2011, 97: 475483. 10.1007/s0001301103244
 13.
Qi XG: Value distribution and uniqueness of difference polynomials and entire solutions of difference equations. Ann. Pol. Math. 2011, 102: 129142. 10.4064/ap10223
 14.
Qi XG, Liu K: Uniqueness and value distribution of differences of entire functions. J. Math. Anal. Appl. 2011, 379: 180187. 10.1016/j.jmaa.2010.12.031
 15.
Ozawa M: On the existence of prime periodic entire functions. Kodai Math. Semin. Rep. 1978, 29: 308321. 10.2996/kmj/1138833654
 16.
Halburd, RG, Korhonen, RJ, Tohge, K: Holomorphic curves with shiftinvariant hyperplane preimages. arXiv:0903.3236
 17.
Clunie J: On integral and meromorphic functions. J. Lond. Math. Soc. 1962, 37: 1727. 10.1112/jlms/s137.1.17
 18.
Yang CC, Laine I: On analogies between nonlinear difference and differential equations. Proc. Jpn. Acad., Ser. A, Math. Sci. 2010, 86: 1014. 10.3792/pjaa.86.10
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11226091, No. 11171013).
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both 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
 uniqueness
 meromorphic functions
 difference polynomials