Some results on zeros and the uniqueness of one certain type of high difference polynomials

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R E S E A R C H

Open Access

Some results on zeros and the uniqueness of

one certain type of high difference

polynomials

Jie Zhang

*

*_{Correspondence:}

[email protected] College of Science, China University of Mining and Technology, Xuzhou, 221116, P.R. China

Abstract

In this paper, we investigate the distribution of the zeros of some high

diﬀerential-diﬀerence polynomials and obtain some uniqueness theorems with regard to it.

MSC: 30D35; 34M10

Keywords: uniqueness; entire function; diﬀerence equation; small function

1 Introduction and main results

In this paper, a meromorphic function always means it is meromorphic in the whole com-plex planeC. We assume that the reader is familiar with the standard notations in the Nevanlinna theory. We use the following standard notations in value distribution theory (see [, , , ]):

T(r,f),m(r,f),N(r,f),N(r,f), . . . .

And we denote byS(r,f) any quantity satisfying

S(r,f) =oT(r,f), asr→ ∞,

possibly outside of a setEwith finite linear measure, not necessarily the same at each occurrence. A meromorphic functiona(z) is said to be a small function with respect tof(z) ifT(r,a) =S(r,f). We say that two meromorphic functionsf(z) andg(z) share the value aIM (ignoring multiplicities) iff(z) –aandg(z) –ahave the same zeros. Iff(z) –aand g(z) –ahave the same zeros with the same multiplicities, then we say that they share the valueaCM (counting multiplicities). A polynomialQ(z,f) is called a differential-difference polynomial inf ifQ is a polynomial inf, its derivatives and shifts with meromorphic coefficients, say{aλ|λ∈I}, such thatT(r,aλ) =S(r,f) for allλ∈I. We define the difference operatorsf =f(z+ ) –f(z) andnf =n–(f).

In , Laine and Yang [] considered zeros of one certain type of diﬀerence polyno-mials and obtained the following theorem.

Theorem A Let f be a transcendental entire function of ﬁnite order and c be a nonzero complex constant. If n≥, then fn₍_z₎_f₍_z₊_c_{) –}_{a has inﬁnitely many zeros, where a}_∈_C_\__.

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Liu [] considered the case of general difference products of a meromorphic function and made relevant improvements to Theorem A. Recently, a number of papers [, ] fo-cusing on the distribution of zeros of some difference polynomials of differential types emerged. In this paper, we consider the general differential difference cases in some sense and obtain some results as follows.

Theorem  Let f be a transcendental entire function of ﬁnite order andα(z)≡be a small function with respect to f(z). If one of the following conditions holds:

(i) n≥k+ whenm≤k+ ; (ii) n≥k–m+ whenm≥k+ ,

then the differential-difference polynomial[fn₍_z₎_fm₍_z_{+ )]}(k)_–_α₍_z₎_{has infinitely many}

ze-ros.

Theorem  Let f be a transcendental entire function of finite order andm_f _≡__{, and} α(z)≡be a small function with respect to f(z). If n≥k+ , then the differential-difference polynomial[fn₍_z₎m_f_](k)_–_α₍_z₎_{has infinitely many zeros.}

Theorem  Let f be a transcendental meromorphic function of finite order,λbe a nonzero constant, andα(z)≡be a small function with respect to f(z). If n≥m+ , then the differential-difference polynomial fn₍_z₎₍_fm₍_z_{+ ) +}_λ_{) –}_α₍_z₎_{has infinitely many zeros.}

Theorem  Let f , g be two transcendental entire functions of ﬁnite order,λbe a nonzero constant, and α(z)≡ be a small function with respect to f(z). If n≥ m+  and fn₍_z₎₍_fm₍_z_{+ ) +}_λ₎_{, g}n₍_z₎₍_gm₍_z_{+ ) +}_λ₎_share_α₍_z₎_{CM, then one of the following cases holds:}

(i) f =hg, wherehn₌_hm_{= , i.e., if}_m_,_n_{are two prime integers, then}_f_≡_g_; (ii) fn₍_z₎₍_fm₍_z_{+ ) +}_λ₎_gn₍_z₎₍_gm₍_z_{+ ) +}_λ_{) =}_α₍_z_).

Recently, Yang and Laine [] considered the following interesting diﬀerential equation and they proved

Theorem B Let p(z)be a nonvanishing polynomial and b,c∈C be two nonzero complex constants. If p(z)is nonconstant, then the diﬀerential equation

f(z) +p(z)f (z) =csinbz

admits no transcendental entire solutions. If p(z)is a nonzero constant, then equation above

admits three distinct transcendental entire solutions, provided(pb_)=c

.

They also presented some results on diﬀerence analogues of the equation in Theorem B and obtained the following theorem.

Theorem C A nonlinear diﬀerence equation

f(z) +q(z)f(z+ ) =csinbz,

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diﬀer-ence equation above admits three distinct transcendental entire functions of ﬁnite order, provided b= nπand q_{= (–)}n+ c

 for a nonzero integer n.

In this paper, we also consider a more general class of diﬀerence equation related to Theorem B and Theorem C and obtain the following results, which generalize the above related results.

Theorem  Let p, q be two polynomials and m≥, n be two positive integers. Then the diﬀerence equation

fm(z) +q(z)nf =p(z)

has no transcendental entire function of ﬁnite order.

Theorem  Let q(z)be a nonconstant polynomial and b, c be two nonzero constants. Then the diﬀerence equation

f₍_z_{) +}_q₍_z₎m_f₌_c_sin_bz _()

has no transcendental entire function of finite order. If q(z)is a nonzero constant, and the difference equation above admits a transcendental entire function of finite order, then

b=kπ

m + π+ nπ

and

q= –c



(–ekmπi– )

m,

where n∈N , k= , , . . . ,m– .

2 Some lemmas

To prove our results, we need some lemmas as follows.

Lemma (see []) Let f(z)be a transcendental meromorphic function with ﬁnite order. Then

m

r,f(z+c) f(z)

=S(r,f).

Lemma (see []) Let f(z)be a transcendental meromorphic function with ﬁnite orderσ

andηbe a nonzero complex number, then for eachε> , we have

Tr,f(z+η)=T(r,f) +Orσ–+ε+O(logr), i.e.,

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Lemma  Let f(z)be a transcendental entire function with ﬁnite order and F=fn₍_z₎_× fm₍_z_{+ )}_{. Then}

Tr,F(z)= (m+n)T(r,f) +S(r,f).

Proof On the one hand, from Lemma , Lemma , and the standard Valiant-Mohon’ko theorem, we obtain

on the other hand, by Lemma  once again, we obtain

T(r,F)≤nTr,f(z)+mTr,f(z+c)+S(r,f)≤(m+n)T(r,f) +S(r,f).

Combing the two equations above, we can get our conclusion immediately.

Lemma  Let f(z)be a transcendental entire function with ﬁnite order andmf ≡. Then

Tr,fn(z)mf≥nT(r,f) +S(r,f).

Proof It is obvious that the diﬀerence operatormf can be expressed in the following form:

Then from the standard Valiant-Mohon’ko theorem and the equation above, we can obtain

Tr,fnmf=T

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Lemma (see []) Let f(z)be a transcendental meromorphic solution of ﬁnite orderσ of a diﬀerence equation of the form

H(z,f)P(z,f) =Q(z,f),

where H(z,f), P(z,f), Q(z,f)are diﬀerence polynomials in f(z)such that the total degree of H(z,f)in f(z)and its shifts is n and that the corresponding total degree of Q(z,f)is at most n. If H(z,f)just contains one term of maximal total degree, then for anyε> ,

mr,P(z,f)=Orσ–+ε+S(r,f)

holds possibly outside of an exceptional set of ﬁnite logarithmic measure.

Lemma (see []) Suppose c is a nonzero constant andαis a nonconstant meromorphic function. Then the diﬀerential equation

f+cf(n)

=α

has no transcendental meromorphic solutions satisfying T(r,α) =S(r,f).

Lemma (see []) Suppose that f(z),f(z), . . . ,fn(z)(n≥) are meromorphic functions and g(z),g(z), . . . ,gn(z)are entire functions satisfying the following conditions:

(i) n_j_=fj(z)egj(z)≡_;

(ii) gj(z) –gk(z)are not constants for≤j<k≤n;

(iii) For≤j≤n,≤h<k≤n.T(r,fj) =o{T(r,egh–gk₎}₍_r→ ∞_,_r∈_/_E_).

Then fj(z)≡(j= , , . . . ,n).

Lemma (see []) Let f be a nonconstant meromorphic function and p, k be positive in-tegers. Then

Np

r,  f(k)

≤Tr,f(k)–T(r,f) +Np+k

r, f

+S(r,f).

Remark Np(r,_f) denotes the counting function of zeros off where an m-fold zero is countedmtimes ifm≤pandptimes ifm>p.

Lemma (see []) Suppose F and G are two nonconstant meromorphic functions. If F and G shareCM, then one of the following three cases holds:

(i) max{T(r,F),T(r,G)} ≤N(r,_G) +N(r,G) +N(r,_F) +N(r,F) +S(r,F) +S(r,G);

(ii) F≡G; (iii) F·G≡.

3 The proof of theorems

The proof of Theorem  Denote F(z) =fn₍_z₎_fm₍_z _{+ ). From the deﬁnition of} _F _and Lemma , we obtain

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which meansFis also a transcendental entire function. On the contrary, we suppose that

Applying the second main theorem for three small functions to the entire functionF(k)_,

we can obtain

And by applying Lemma  to the right side of the equation above, we can obtain

Tr,F(k)≤Tr,F(k)–T(r,F) +Nk+

Combing Equations ()-(), we obtain

(n+m)T(r,f) +S(r,f) =T(r,F)≤Nk+

Next, we discuss the following two cases separately.

Case . Ifm≤k+ , then from the deﬁnition ofFagain, we can obtain

Then, based on the equation above, Equation () becomes

(n–k– )T(r,f)≤S(r,f),

which contradicts our assumptionn≥k+ .

Case . Ifm≥k+ , then in the similar way as in Case , we can obtain

Then, based on the equation above, Equation () becomes

(n+m– k– )T(r,f)≤S(r,f),

which also contradicts our assumptionn≥k+  –m. Therefore,F(k)_–_α₍_z_{) has inﬁnitely}

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The proof of Theorem  DenoteF(z) =fn₍_z₎m_f_{. From the deﬁnition of}_F_{and Lemma ,} we get

T(r,F)≥nT(r,f) +S(r,f),

which meansF is also a transcendental entire function. On the contrary, suppose that F(k)_–_α₍_z_{) has only ﬁnitely many zeros. Noting the following fact in Lemma  again}

Then in the similar way as in Theorem , we can obtain

T(r,F)≤Nk+

zeros. The proof of Theorem  is completed.

The proof of Theorem  DenoteF(z) =fn₍_z₎₍_fm₍_z_{+ ) +}_λ_{). On the contrary, suppose that} F(z) –α(z) has only ﬁnitely many zeros. Then from the deﬁnition of F, we can obtain obviously that

T(r,F)≥(n–m)T(r,f) +S(r,f),

which meansFis also a transcendental entire function. Applying the second main theorem for three small functions to functionF, we obtain

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In other words,

(n– m– )T(r,f)≤S(r,f),

which contradicts our assumptionn≥m+ . So we provedF(z) –α(z) has inﬁnitely many

zeros. The proof of Theorem  is completed.

The proof of Theorem  Set

F=f

n₍_z₎₍_fm₍_z_{+ ) +}_λ₎

α(z) and G=

gn(z)(gm(z+ ) +λ)

α(z) .

ThenFandGshare  CM. From the deﬁnition ofF,Gand Lemma , we obtain

(n–m)T(r,f) +S(r,f)≤T(r,F)≤(n+m)T(r,f) +S(r,f) ()

and

(n–m)T(r,g) +S(r,g)≤T(r,G)≤(n+m)T(r,g) +S(r,g). ()

From the deﬁnition ofN(r,_F) and Lemma , we obtain

N

r, F

≤N

r, f

+N

r, 

fm₍_z_{+ ) +}_λ

≤T(r,f) +mTr,f(z+ )+S(r,f)

≤( +m)T(r,f) +S(r,f).

In other words,

N

r, F

+N(r,F)≤( +m)T(r,f) +S(r,f). ()

In the similar way, we can obtain

N

r,  G

+N(r,G)≤( +m)T(r,g) +S(r,g). ()

Next, we consider three cases according to Lemma .

Case . Suppose the ﬁrst equation in Lemma  holds. Then by Equations ()-(), we obtain

maxT(r,F),T(r,G)≤( +m)T(r,f) +T(r,g)+S(r,f) +S(r,g). ()

But by Equations (), (), and (), we obtain

(n– m– )maxT(r,F),T(r,G)≤S(r,f) +S(r,g), ()

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Case . SupposeF≡Gholds. That is to say

fn(z)fm(z+ ) +λ=gn(z)gm(z+ ) +λ. ()

Seth=f_g, and suppose it is a nonconstant meromorphic function. Ifhn₍_z₎_hm₍_z_{+ )}_≡_{, then}_hn₍_z_{) =} 

hm₍_z_+). By Lemma , we obtain

nTr,h(z)=mTr,h(z+ )+O() =mTr,h(z)+S(r,f),

which is also impossible whenn≥m+ . Sohn₍_z₎_hm₍_z_{+ )}_≡_. Substitutingh=f_g into Equation (), we obtain

gm(z+ ) =λ  –h n

hn₍_z₎_hm₍_z_{+ ) – }. ()

From Equation (), it is obvious that if hn(z)hm(z+ ) –  = , then  –hn(z) =  and hm₍_z_{+ ) –  = . Thus, we can obtain}

N

r, 

hn₍_z₎_hm₍_z_{+ ) – }

≤N

r, 

hm₍_z_{+ ) – }

≤mT(r,h) +S(r,h).

Applying the second main theorem toH:=hn₍_z₎_hm₍_z_{+ ) and noting the equation above,} we obtain

(n–m)T(r,h) +S(r,h)

≤T(r,H)≤N

r,  H

+N

r,  H– 

+N(r,H) +S(r,H)

≤N(r,h) +Nr,h(z+ )+N

r, h

+N

r,  h(z+ )

+mT(r,h) +S(r,h)

≤( +m)T(r,h) +S(r,h).

That is to say,

(n– m– )T(r,h) =S(r,h),

which is impossible whenn≥m+ . Thus, we obtainhis a constant such thathn₌_hm_{= .} Furthermore, if (m,n) = , thenh=  andf ≡g.

Case . SupposeF·G≡ holds. Then

fn(z)fm(z+ ) +λ·gn(z)gm(z+ ) +λ=α(z).

The proof of Theorem  is completed.

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Then, by Lemma , we obtain

mT(r,f) =mm(r,f) =mr,p(z) –q(z)nf≤mr,nf+S(r,f)

=m(r,f) +S(r,f) =T(r,f) +S(r,f),

which is impossible whenm> . The proof of Theorem  is completed.

The Proof of Theorem  Suppose the diﬀerence equation () admits a transcendental entire function of ﬁnite order. First of all, we notice that

mf =f(z+m) +αm–f(z+m– ) +· · ·+αf(z+ ) + (–)m+f(z)

is a diﬀerence polynomial off with total degree at most one. Diﬀerentiating Equation (), we obtain

ff +qQ+qQ =bccosbz, ()

whereQ=mf. Combining the squares of Equation () and Equation (), we can obtain

f bf+ f =Q(f), ()

where

Q(f) =  –bqQ– bqQf–

qQ +qQ– ff qQ +qQ

is a diﬀerential-diﬀerence polynomial of f with total degree at most four. Then by Lemma , we obtain

Tr,bf+ f =mr,bf+ f =S(r,f). ()

From Equation () and Lemma , we obtainb_f_{+ }_f 

must be a constant. To put it another way,

bf+ f = . ()

It is clear that the solutions of homogeneous and linear Equation () with constant coef-ﬁcients must be of the form

f(z) =ce

biz  ₊_c

e

–biz

 _, _()

wherec,care two constants. Next, we claimc,care two nonzero constants.

Ifc= , thenf(z) =ce

biz

 _and_f₌_Af_,m_f ₌_Am_f_{, where}_A₌_ebi _{– . Substituting these}

into Equation (), we obtain

c_ebiz+cAmq(z)e

biz  ₌ c

i

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By Lemma , we obtainc=c= , which is impossible. Thusc= , and we can obtain

c=  in a similar way. Thus, our claim holds.

From Equation (), we obtain

f =cAe

By Lemma  again, we obtain

c_= c

Ifq(z) is a nonconstant polynomial, then we can obtain a contradiction from Equation () obviously.

wherenis an integer. From Equation (), we obtain

q= –c



(–ekmπi_{– )}

m.

The proof of Theorem  is completed.

Competing interests

The author declares that they have no competing interests.

Authors’ contributions

The author carried out the proof of the theorems and approved the ﬁnal manuscript.

Acknowledgements

The author would like to thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This research was supported by the Fundamental Research Funds for the Central Universities (No. 2011QNA25).

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doi:10.1186/1687-1847-2012-160