We’d like to understand how you use our websites in order to improve them. Register your interest.

# Meromorphic functions sharing small functions with their linear difference polynomials

## 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 ). For any given nonconstant meromorphic function $f(z)$, we recall the hyper order of $f(z)$ defined as follows (see ):

$ρ 2 (f):= lim sup r → ∞ log log T ( r , f ) log r .$

Denote by $S(r,f)$ any quantity satisfying $S(r,f)=o(T(r,f))$ as $r→∞$, 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 $a∈S(f)∪S(g)∪{∞}$, 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 $c∈C$, $f(z+c)$ is called a shift of $f(z)$. And a difference monomial of type $∏ i = 1 m f n i (z+ c i )$ is called a difference product of $f(z)$, where $c 1 ,…, c m ∈C$ and $n 1 ,…, n m ∈N$.

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

$L(z,f)= ∑ i = 1 n a i (z)f(z+ c i ),$

where $c 1 ,…, c n ∈C$, $a 1 (z),…, a n (z)∈S(f)$.

It is well known that the difference operators of $f(z)$ are defined as follows:

$Δ c f(z)=f(z+c)−f(z)and Δ c n f(z)= Δ c n − 1 ( Δ c f ( z ) ) ,n∈N,n≥2.$

In particular, $Δ c n f(z)= Δ n f(z)$ 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 $f ′ (z)$ was initiated by Rubel and Yang . For a nonconstant entire function $f(z)$, they proved that $f(z)≡ f ′ (z)$ provided that $f(z)$ and $f ′ (z)$ 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., ) or difference operators (see, e.g., [9, 12]). The following Theorem A is indeed a corollary of Theorem 2.1 in  and Theorem 2 in .

Theorem A ([10, 11])

Let $f(z)$ be a meromorphic function of finite order, let $c∈C$, and let $a 1 , a 2 ∈S(f)$ 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 $z∈C$.

Theorem B below is Theorem 1.2 in , while Theorem C is Theorem 1.1 in .

Theorem B ()

Let $f(z)$ be a transcendental meromorphic function such that its order of growth $ρ(f)$ is not an integer or infinite, and let $c∈C$ be a constant such that $f(z+c)≢f(z)$. If $Δ c f(z)$ and $f(z)$ share three distinct values a, b, ∞ CM, then $f(z+c)=2f(z)$.

Theorem C ()

Let $f(z)$ be a nonconstant entire function of finite order, $c∈C$, and n be a positive integer. Suppose that $f(z)$ and $Δ c n f(z)$ share two distinct finite values a, b CM and one of the following cases is satisfied:

1. (i)

$ab=0$;

2. (ii)

$ab≠0$ and $ρ(f)∉N$.

Then $f(z)≡ Δ c n f(z)$.

Remark 1 The methods in  and  are quite different. Due to a result of Ozawa  (he proved that for any given $ρ∈[1,∞)$, there exists a periodic entire function of order ρ), Chen and Yi  and Li and Gao  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  and , we obtain the following result with a quite simple proof.

Theorem 1.1 Let $f(z)$ be a meromorphic function of hyper order $ρ 2 (f)<1$, let $L(z,f)$ be a difference polynomial of $f(z)$, and let $a,b∈S(f)$ be two distinct meromorphic functions. Suppose that $f(z)$ and $L(z,f)$ share a, b, ∞ CM and one of the following cases holds:

1. (i)

$L(z,a)−a=L(z,b)−b≡0$;

2. (ii)

$L(z,a)−a≡0$ or $L(z,b)−b≡0$, and $N(r,f)<λT(r,f)$ for some $λ∈(0,1)$;

3. (iii)

$ρ(f)∉N∪{∞}$.

Then $f(z)≡L(z,f)$.

Example 1 We give two examples for Theorem 1.1.

1. (1)

For the cases (i) and (ii): Let $f(z)= e z log 3$ and $L(z,f)=Δf(z)−f(z)=f(z+1)−2f(z)$. Then $L(z,f)=f(z)$, and hence for $a=0$ and any given $b∈S(f)$, $f(z)$ and $L(z,f)$ share a, b, ∞ CM.

2. (2)

For the case (iii): Let $f(z)=g(z) e z log 3$ and $L(z,f)=Δf(z)−f(z)=f(z+1)−2f(z)$, where $g(z)$ is a periodic entire function with period 1 such that $ρ(g)∈(1,∞)∖N$. Then $L(z,f)=f(z)$, and hence for any given $a,b∈S(f)$, $f(z)$ and $L(z,f)$ share a, b, ∞ CM.

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

Theorem D ()

Let $f(z)$ be a nonconstant entire function of finite order $ρ(f)$, $η∈C$. If $f(z)$ and $f(z+η)$ share one finite value a CM, and for a finite value $b≠a$, $f(z)−b$ and $f(z+η)−b$ have $max{1,[ρ(f)]−1}$ distinct common zeros of multiplicity ≥2, then $f(z)≡f(z+η)$.

Theorem E ()

Let $f(z)$ be a nonconstant entire function of finite order $ρ(f)$, $η∈C$, and n be a positive integer. If $f(z)$ and $Δ η n f(z)$ share one finite value a CM, and for a finite value $b≠a$, $f(z)−b$ and $Δ η n f(z)−b$ have $max{1,[ρ(f)]}$ distinct common zeros of multiplicity ≥2, then $f(z)≡ Δ η n f(z)$.

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 $ρ(f)$, let $L(z,f)$ be a difference polynomial of $f(z)$, and let $a,b∈S(f)$ 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 $m=max{1,[ρ(f)]}$ distinct common zeros of multiplicity ≥2, denoted by $z 1 , z 2 ,…, z m$, such that $a( z i )≠b( z i )$. Then $f(z)≡L(z,f)$.

Example 2 Let $f(z)= g 2 (z) e z log 3$ and $L(z,f)=Δf(z)−f(z)=f(z+1)−2f(z)$, where $g(z)$ is a periodic entire function with period 1 such that $ρ(g)∈(1,∞)∖N$. Then $L(z,f)=f(z)$, and hence for any given $a∈S(f)$ and $b=0$, $f(z)$ and $L(z,f)$ share a, ∞ CM.

Remark 2 Chen and Yi  (resp. Li and Gao ) 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 ()

Let $f(z)$ be a meromorphic function of hyper order $ρ 2 (f)=ς<1$, $c∈C$, and $ε>0$. Then

$m ( r , f ( z + c ) f ( z ) ) =o ( T ( r , f ) r 1 − ς − ε ) ) =S(r,f),$

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

The following lemma is a Clunie-type lemma  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  and stated as follows.

Lemma 2.2 ()

Let $f(z)$ be a meromorphic function of hyper order $ρ 2 (f)<1$ and $P(z,f)$, $Q(z,f)$ be two difference-differential polynomials of f. If

$f n P(z,f)=Q(z,f)$

holds and if the total degree of $Q(z,f)$ in f and its derivatives and their shifts isn, 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

$L − a f − a = e p$
(3.1)

and

$L − b f − b = e q ,$
(3.2)

where $p=p(z)$, $q=q(z)$ are entire functions such that $max{ρ( e p ),ρ( e q )}≤ ρ 2 (f)$.

It follows from (3.1) and (3.2) that

$( e q − e p ) f=a−b+b e q −a e p .$
(3.3)

If $e p ≡ e q$, then from (3.3) we obtain

$(a−b) ( 1 − e p ) =0.$

Since $a−b≢0$, we get $e p ≡1$ and hence finish our proof from (3.1).

Next, we assume that $e p ≢ e q$ and complete our proof in three steps.

Step 1. We prove the case (i): $L(z,a)−a=L(z,b)−b≡0$. From (3.1), we get from Lemma 2.1 that

$T ( r , e p ) =m ( r , e p ) =m ( r , L − a f − a ) =m ( r , L ( z , f − a ) f − a ) =S(r,f).$
(3.4)

Similarly, we have $T(r, e q )=S(r,f)$. Now, we can deduce a contradiction from (3.3) that

$T ( r , f ) = T ( r , a − b + b e q − a e p e q − e p ) ≤ 2 ( T ( r , a ) + T ( r , b ) + T ( r , e p ) + T ( r , e p ) ) + S ( r , f ) = S ( r , f ) .$

Step 2. We prove the case (iii): $ρ(f)∉N∪{∞}$. In this case, $p(z)$, $q(z)$ are polynomials, and we have

$max { deg p ( z ) , deg q ( z ) } ≤ [ ρ ( f ) ] <ρ(f).$
(3.5)

From (3.3), we obtain

$T(r,f)=T ( r , a − b + b e q − a e p e q − e p ) ≤2T ( r , e p ) +2T ( r , e q ) +S(r,f),$

which gives $ρ(f)≤max{degp(z),degq(z)}$, a contradiction to (3.5).

Step 3. We prove the case (ii): $L(z,a)−a≡0$ or $L(z,b)−b≡0$, and $N(r,f)<λT(r,f)$ for some $λ∈(0,1)$. Without loss of generality, we assume that $L(z,a)−a≡0$ and hence (3.4) still holds.

Differentiating (3.1) and (3.2), we get

$L ′ f− f ′ L− p ′ fL=a ( L ′ − f ′ ) + a ′ (f−L)−a p ′ (f+L)+ p ′ a 2$

and

$L ′ f− f ′ L− q ′ fL=b ( L ′ − f ′ ) + b ′ (f−L)−b q ′ (f+L)+ p ′ b 2 .$

Combining two equations above, we get

$A 1 fL= A 2 ( L ′ − f ′ ) + A 3 f+ A 4 L+ A 5 ,$
(3.6)

where $A 1 = p ′ − q ′$, $A 2 =b−a$, $A 3 = b ′ − a ′ +a p ′ +b q ′$, $A 4 = a ′ − b a ′ p ′ +b q ′$, $A 5 = q ′ b 2 − p ′ a 2$.

Notice that the right-hand side of (3.6) is a difference-differential 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

$m(r,f)=m ( r , L − a e p + a ) ≤m ( r , e p ) +m(r,L)+2m(r,a)+S(r,f)=S(r,f),$

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

## 4 Proof of Theorem 1.2

Since $f(z)$ and $L(z,f)$ share a CM, we have

$L ( z , f ) − a f ( z ) − a = e p ,$
(4.1)

where p is a polynomial such that $degp(z)≤max{1,[ρ(f)]}=m$.

It follows from (4.1) that

$L ′ (z,f)− a ′ = ( f ′ ( z ) − a ′ ) e p + p ′ ( f ( z ) − a ) e p .$
(4.2)

For each point $z i$, $1≤i≤m$, satisfying the assumption in Theorem 1.2, we get

$f( z i )=L( z i ,f)=b( z i )≠a( z i ),$
(4.3)

and

$f ′ ( z i )− b ′ ( z i )= L ′ ( z i ,f)− b ′ ( z i )=0,$
(4.4)

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 $p ′ ( z i )=0$. By assumption, $p ′ (z)$ has at least m zeros. This means that $p ′ (z)≡0$. Therefore,

$L(z,f)−a=c ( f ( z ) − a )$
(4.5)

holds for some nonconstant c. For the point $z 1$ such that $L( z 1 ,f)=f( z 1 )=b( z 1 )≠a( z 1 )$, we get from (4.5) that $c=1$ and hence prove that $f(z)≡L(z,f)$.

## References

1. 1.

Hayman WK: Meromorphic Functions. Clarendon Press, Oxford; 1964.

2. 2.

Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.

3. 3.

Yang CC, Yi HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.

4. 4.

Rubel LA, Yang CC Lecture Notes in Math. 599. In Values Shared by an Entire Function and Its Derivative. Springer, Berlin; 1977:101-103.

5. 5.

Chiang YM, Feng SJ:On the Nevanlinna characteristic $f(z+η)$ and difference equations in the complex plane. Ramanujan J. 2008, 16: 105-129. 10.1007/s11139-007-9101-1

6. 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: 3767-3791. 10.1090/S0002-9947-09-04663-7

7. 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: 477-487. 10.1016/j.jmaa.2005.04.010

8. 8.

Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463-478.

9. 9.

Chen ZX, Yi HX: On sharing values of meromorphic functions and their differences. Results Math. 2013, 63: 557-565. 10.1007/s00025-011-0217-7

10. 10.

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. 10.1080/17476930903394770

11. 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: 352-363. 10.1016/j.jmaa.2009.01.053

12. 12.

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/s00013-011-0324-4

13. 13.

Qi XG: Value distribution and uniqueness of difference polynomials and entire solutions of difference equations. Ann. Pol. Math. 2011, 102: 129-142. 10.4064/ap102-2-3

14. 14.

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.2010.12.031

15. 15.

Ozawa M: On the existence of prime periodic entire functions. Kodai Math. Semin. Rep. 1978, 29: 308-321. 10.2996/kmj/1138833654

16. 16.

Halburd, RG, Korhonen, RJ, Tohge, K: Holomorphic curves with shift-invariant hyperplane preimages. arXiv:0903.3236

17. 17.

Clunie J: On integral and meromorphic functions. J. Lond. Math. Soc. 1962, 37: 17-27. 10.1112/jlms/s1-37.1.17

18. 18.

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.86.10

## Acknowledgements

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

## Author information

Authors

### Corresponding author

Correspondence to BaoQin Chen.

### 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

Reprints and Permissions

Li, S., Chen, B. Meromorphic functions sharing small functions with their linear difference polynomials. Adv Differ Equ 2013, 58 (2013). https://doi.org/10.1186/1687-1847-2013-58 