Entire functions sharing a small function with their two difference operators

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R E S E A R C H

Open Access

Entire functions sharing a small function

with their two difference operators

Feng Lü

1

_{, Yanfeng Wang}

1

_{and Junfeng Xu}

2*

*_{Correspondence:}

[email protected]

2_{Department of Mathematics, Wuyi}

University, Jiangmen, Guangdong 529020, P.R. China

Full list of author information is available at the end of the article

Abstract

In this article, we deduce a uniqueness result of entire functions that share a small entire function with their two diﬀerence operators, generalizing some previous theorems of (Farissi et al. in Complex Anal. Oper. Theory 10:1317-1327, 2015, Theorem 1.1) and (Chen and Li in Adv. Diﬀer. Equ. 2014:311, 2014, Theorem 1.1) by omitting the assumption that the shared small entire function is periodic.

MSC: 30D35; 30D30; 39A10

Keywords: uniqueness; entire functions; diﬀerence operators; Nevanlinna theory

1 Introduction and main result

Nevanlinna theory of value distributions is concerned with the density of points where a meromorphic function takes a certain value in the complex plane. Nowadays, there has been recent interest in connections between the Nevanlinna theory and the diﬀerence op-erator. In addition, many papers have been devoted to the investigation of the uniqueness problems related to meromorphic functions and their shifts or their diﬀerence operators and one got a lot of results (see,e.g., [–]).

In order to state the main result, we give the following definition. For a meromorphic functionf(z), we define its shift byfc=f(z+c) and its difference operators by

cf(z) =f(z+c) –f(z),

_cnf(z) =_cn–cf(z)

, n∈N,n≥.

A meromorphic functiona(z) is said to be a small function with respect tof(z) if and only ifT(r,a) =S(r,f), whereS(r,f) =o(T(r,f)), asr→ ∞outside of a possible exceptional set of ﬁnite logarithmic measure. Denote the set of all the small functions off(z) byS(f). Let

f(z) andg(z) be two meromorphic functions and leta(z) be a small entire function off(z) andg(z). We say thatf(z) andg(z) sharea(z) IM, provided thatf(z) –a(z) andg(z) –a(z) have the same zeros ignoring multiplicities. Similarly, we say thatf(z) andg(z) sharea(z) CM, provided thatf(z) –a(z) andg(z) –a(z) have the same zeros counting multiplicities. Recently, Chenet al.[, ] investigated two uniqueness problems on entire functions that share a small periodic entire function with their two diﬀerence operators as follows.

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Theorem A(see [], Theorem .) Let f(z)be a nonconstant entire function of ﬁnite order,

let a(z)(≡)∈S(f)be a periodic entire function with period c.If f(z),cf(z),cf(z)share

a(z)CM,then_cf(z)≡cf(z).

Theorem B(see [], Theorem .) Let f(z)be a nonconstant entire function of ﬁnite order.

If f(z),cf(z),cf(z)shareCM,thencf(z)≡Ccf(z),where C is a nonzero constant.

In , El Farissi, Latreuch and Asiri further studied the above problem and obtained

Theorem C(see [], Theorem .) Let f(z)be a nonconstant entire function of ﬁnite order,

let a(z)(≡)∈S(f)be a periodic entire function with period c.If f(z),cf(z),cf(z)share

a(z)CM,then f(z)≡cf(z).

Remark  It is necessary to point out that Theorems A and B have been generalized from

_cf(z) to n_cf(z) by Chen, Chen and Li in []. There are also some interesting results related the above theorems (see,e.g., [, ]).

In the previous results, we ﬁnd that the shared small functiona(z) is a periodic function with periodc. So, it is natural to ask what will happen if the periodic condition ofa(z) is omitted. In this paper, we focus on this problem and we obtain the following result.

Theorem  Let f(z)be a nonconstant entire function of ﬁnite order,and let a(z)∈S(f)be an entire function.If f(z),cf(z),cf(z)share a(z)CM,then one of the following assertions

holds:

(i) Ifca(z)≡a(z),thencf(z) –a(z) =C(cf(z) –a(z)),whereCis a nonzero

constant.

(ii) Ifca(z)≡a(z),thencf(z) =f(z)orcf(z) –a(z) =eγ(cf(z) –a(z)),whereγ is a

polynomial withdegγ <ρ(a).

Remark  We point out that Theorem  is a generalization of the previous theorems.

If a(z)≡, then ca(z) =a(z). Then it follows from (i) of Theorem  thatcf(z) =

Ccf(z), whereCis a nonzero constant.

Ifa(z)≡ is a periodic function with periodc, thenca(z)≡a(z). It follows from (ii)

of Theorem  thatcf(z) =f(z) orcf(z) =cf(z). Furthermore, by Theorem C we can

deduce thatcf(z) =f(z).

As an application of Theorem , we can obtain an interesting result, wherea(z) is a slow growth small function.

Theorem  Let f(z)be a nonconstant entire function of ﬁnite order,and let a(z)(≡)∈S(f)

be an entire function withρ(a) < .If f(z),cf(z),cf(z)share a(z)CM,thencf(z) =f(z).

For convenience of the reader, we list here some notations. For a meromophic function

f, we use the basic notations of the Nevanlinna theory of meromorphic functions such as

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2 Some lemmas

In this section, we state some results that we employ in our proofs.

Lemma .([], Theorem .) Let c∈C,n∈N,and let f be a meromorphic function with a ﬁnite order.Then for all small periodic functions a(z)∈S(f)

m

r,

n cf(z)

f(z) –a(z)

=S(r,f),

where S(r,f) =o(T(r,f))for all r outside of a possible exceptional set E with ﬁnite logarith-mic measure.

Lemma .([], Lemma .) Let g be a nonconstant meromorphic function in the plane of order less than,and let h> .Then there exists a-set E such that

g(z+η)

g(z) →, as z→ ∞inC\E,

uniformly inηfor|η|<h.

Lemma . plays an important role in the proof of Theorem .

3 Proof of Theorem 1

Note thatf(z) is a nonconstant entire function of ﬁnite order. Thencf(z) andcf(z) are

also two entire functions of ﬁnite order. Setg(z) =f(z) –a(z). Then

cg(z) =cf(z) –ca(z), cg(z) =cf(z) –ca(z).

Sincef(z),cf(z),cf(z) sharea(z) CM, we have

cf(z) –a(z)

f(z) –a(z) =

cg(z) +ca(z) –a(z)

g(z) =e

P(z)_, _() _cf(z) –a(z)

f(z) –a(z) =

_cg(z) +_ca(z) –a(z)

g(z) =e

Q(z)_, _()

whereP(z) andQ(z) are two polynomials.

Suppose thatca(z)≡a(z). Obviously, we can getca(z)≡a(z). It is clear thatg(z),

cg(z), andcg(z) share  CM. By Theorem B, we can obtaincg(z)≡Ccg(z), where

C is a nonconstant. So

cf(z) –ca(z)≡C(cf(z) –ca(z)). That is,cf(z) –a(z)≡

C(cf(z) –a(z)).

In the following, we assume thatca(z)≡a(z). We consider into two cases.

Case ._ca(z)≡a(z). Set

ϕ(z) = (ca(z) –a(z))(



cf(z) –ca(z)) – (ca(z) –a(z))(cf(z) –ca(z))

f(z) –a(z) ()

=(ca(z) –a(z))(



cf(z) –a(z)) – (ca(z) –a(z))(cf(z) –a(z))

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From () and (), we can rewrite the above function as

ϕ(z) =ca(z) –a(z)

eQ(z)–_ca(z) –a(z)eP(z), ()

which implies thatϕis an entire function.

By () and Lemma ., we deduce thatϕ(z)∈S(f).

By the second main theorem, we deduce

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Combining equations () and () yields

f(z) +eP(z+c) +eP(z)–  +eP(z)+ =f(z)eQ(z), ()

a(z+c) –eP(z+c)+a(z) +eP(z+c) –eP(z)–  –eP(z)=a(z) –eQ(z). () From (), we have

eP(z)eP(z+c)+eP(z+c)–P(z)– =eQ(z). ()

Suppose thateP(z)_{is a nonconstant function. Denote}_η₌_eP(z+c)–P(z)_{– . Obviously,}_η_{is a}

small function ofeP(z)_{. It follows from () that}_eP(z+c)₊_η_{have no zeros. Note that}_eP(z+c)

has only one Picard value, say . Thenη≡, which implieseP(z+c)–P(z)_{= . Again by (),}

we have

eP(z)=eQ(z).

IfeP(z)_{is a constant, then}_eP(z+c)₌_eP(z)_{. By () we also get (}_eP(z)₎₌_eQ(z)_.

Furthermore, it follows from (),eP(z+c)₌_eP(z)_{and (}_eP(z)₎₌_eQ(z)_that

a(z+c) – a(z) –eP(z)= ,

ca(z) –a(z)

 –eP(z)= .

Note thata(z)=ca(z), theneP(z)≡, soeQ(z)≡. Hence, we obtaincf(z) =f(z).

Subcase ..We assume thatϕ(z)≡. Then it follows from () and () that



cf(z) –ca(z)

cf(z) –ca(z)

=



ca(z) –a(z)

=eQ–P=eγ,

whereγ =Q–Pis a polynomial anddegγ<ρ(a). From this, we also have_cf(z) –a(z) =

eγ₍

cf(z) –a(z)).

Case ._ca(z)≡a(z).

By () and Lemma ., we geteQ∈S(f). Rewrite () and () as

cg(z) =g(z)eP(z)+a(z) –ca(z), ()

_cg(z) =g(z)eQ(z). () Then

eQ(z)g(z) =eP(z+c)g(z+c) –eP(z)g(z) +ca(z) –a(z). ()

Rewrite () asg(z+c) =g(z)[ +eP(z)_{] +}_a₍_z_{) –}

ca(z).

Now, substitute the form ofg(z+c) into () yields

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Suppose thatn=degP≤degQ. SinceeQ_∈_S₍_f_{), we have}_eP_∈_S₍_f_).

Similar to the discussion of Subcase ., we obtaincf(z) =f(z).

Now, we assume thatdegP>degQ. Suppose that

P(z) =anzn+an–zn–+· · ·+a, an= .

Seth=eanzn_{. Then}_eP(z)₌_β_h_,_eP(z+c)₌_β_h_{, where}_β

,βare two small functions ofh.

Note thatdegP>degQ, soeQ_{is also a small function of}_h_.

From (), we have

ca(z) –a(z)

g(z) =

eP(z)_eP(z+c)₊_eP(z+c)_–_eP(z)_–_eQ(z)

eP(z+c)_{– } . ()

We claim that

eP(z)_eP(z+c)₊_eP(z+c)_–_eP(z)_–_eQ(z) eP(z+c)_{– }

is irreducible except the factor which is a small function ofh.

Suppose thatzis a common zero ofeP(z+c)_{–  and}_eP(z)_eP(z+c)₊_eP(z+c)_–_eP(z)_–_eQ(z)_{. It is}

easy to deduce thateQ(z)_{= . Note that}_eQ(z)_{– }∈_S₍_h_{). Thus, the claim holds.} Rewrite () as

ca(z) –a(z)

g(z) =

ββh+ (β–β)h–eQ(z)

βh–  . ()

DenoteH=ββh+ (β–β)h–eQ(z)=αh+αh+α, whereαj∈S(h),j= , , . By the

above equation, we see thatT(r,g) =O(T(r,h)), which implies thata∈S(h). Next we can prove that

N

r, 

H

≤N

r, 

ca(z) –a(z)

+S(r,h).

In fact, from () we have

H=ca(z) –a(z)

g(z)

eP(z+c)– .

Note thatg(z) is an entire function. All the zeros ofHcome from the zeros ofca(z) –a(z)

andeP(z+c)– . ByνF(z) we denote the multiplicity of the zero of meromorphic functionF

at the pointz.

Suppose thatzis a zero ofH. We claimνH(z)≤νca–a(z) +degP·νeQ_–(z_). Now we

split into two cases.

Case A. zis not a zero ofeP(z+c)– . Thenzmust be a zero ofca–a. HenceνH(z)≤ νca–a(z).

Case B. zis a zero ofeP(z+c)_{– . Set}_P₍_z_{) =}_P₍_z₊_c_{). Obviously,}_ν

eP–(z)≤degP. Then H(z) =  andeP(z+c)_{–  = , which leads to}_eQ(z)_{= .}

Assume thateQ_≡_{. Then we can rewrite () as}

ca(z) –a(z)

g(z) =

eP(z)_eP(z+c)₊_eP(z+c)_–_eP(z)_{– } eP(z+c)_{– } =e

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Further,

a contradiction. Thus, the case cannot occur. Hence, we ﬁnish the proof of Theorem .

4 Proof of Theorem 2

Ifa(z) is a constant, then it follows from Theorem C thatf(z)≡cf(z). In the following,

we assume thata(z) is a nonconstant entire function. Suppose thatca(z)≡a(z). Then we have

a(z+c)

a(z) = .

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Then there exists a-setEof ﬁnite logarithmic measure, so that

a(z+c)

a(z) →,

for allz→ ∞inC\E. It is absurd. Thus,ca(z)≡a(z). It follows from (ii) of Theorem 

thatcf(z) =f(z) or

_cf(z) –_ca(z)

cf(z) –ca(z)

=



ca(z) –a(z)

=eγ,

whereγ is a polynomial. Suppose that

_cf(z) –_ca(z)

cf(z) –ca(z)

=



ca(z) –a(z)

=eγ.

Note thatρ(a) < . We get

ρeγ_≤_ρ



ca(z) –a(z)

≤ρ(a) < ,

which implies thateγ _{is a nonzero constant, say}_C_{. Furthermore, we get}

_ca(z) –a(z) =Cca(z) –a(z)

.

Rewrite it as

a(z+ c) – ( +C)a(z+c) + Ca(z) = .

Applying Lemma . again, we deduce that  – ( +C) + C= , which implies thatC= . Then the above diﬀerence equation reduces to

a(z+ c) – a(z+c) + a(z) = .

Setb(z) =a(z+c) – a(z). Then the above equation can be rewritten as

b(z+ ) =b(z),

which implies that b(z) is a periodic function. If b(z) is a nonconstant function, then

ρ(b)≥. It contradictsρ(b)≤ρ(a) < . Thus,bis a constant. Hence

a(z+c) – a(z) =b. ()

Then applying Lemma . again, there exists a-setEof ﬁnite logarithmic measure, so that

a(z+c)

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for allz→ ∞inC\E. Rewrite () as

a(z+c)

a(z) –  =

b a(z).

Then choose a sequence{zk}such that |zk|=rk,zk∈/ E and|a(zk)|=M(rk,a),rk→ ∞

ask→ ∞. Substitutingzkinto the above function yields a contradiction by Lemma ..

Thus, this case cannot occur. Then we deduce the desired result. Hence, we ﬁnish the proof of Theorem .

Acknowledgements

The research was supported by NNSF of China Project (No. 11601521), the Fundamental Research Fund for Central Universities in China (Nos. 15CX05061A, 15CX05063A and 15CX08011), NSF of Guangdong Province (Nos. 2016A030313002, 2015A030313644) and Funds of Education Department of Guangdong (2016KTSCX145).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{College of Science, China University of Petroleum, Qingdao, Shandong 266580, P.R. China.}2_{Department of Mathematics,}

Wuyi University, Jiangmen, Guangdong 529020, P.R. China.

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Received: 12 December 2016 Accepted: 17 July 2017 References

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