On the value distribution and uniqueness of difference polynomials of meromorphic functions

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R E S E A R C H

Open Access

On the value distribution and uniqueness of

difference polynomials of meromorphic

functions

Hong-Yan Xu

*

*_{Correspondence:}

[email protected]

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

Abstract

In this paper, we study the zeros of diﬀerence polynomials of meromorphic functions of the forms

P(f)

d

j=1

f(z+cj)sj

(k)

–

α

(z),

P(f)

d

j=1

f(z+cj) –f(z)

sj (k)

–

α

(z),

whereP(f) is a nonzero polynomial of degreen,cj∈C\ {0}(j= 1,. . .,d) are distinct

constants,n,k,d,sj(j= 1,. . .,d)∈N+, and

α

(z) is a small function off. Our results of

this paper are an improvement of the previous theorems given by Chen and Chen (Chinese Ann. Math., Ser. A 33(3):359-374, 2012) and Liuet al.(Appl. Math. J. Chin. Univ. Ser. B 27(1):94-104, 2012). In addition, we also investigate the uniqueness for

diﬀerence polynomials of transcendental entire functions sharing one value and obtain some results which extend the recent theorem given by Liuet al.(Adv. Diﬀer. Equ. 2011:234215, 2012).

MSC: 30D35; 39A10

Keywords: uniqueness; meromorphic function; hyper-order

1 Introduction and main results

This purpose of this paper is to study the problem of the zeros and uniqueness of com-plex difference polynomials of meromorphic functions. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions are used (see [, ]). In addition, for a meromorphic functionf, we useS(r,f) to denote any quantity satisfyingS(r,f) =o(T(r,f)) for allroutside a possible exceptional setEof finite logarithmic measurelim_r→∞_[,_r₎_∩Edt_t <∞. A meromorphic functiona(z) is called a small function with respect tof ifT(r,a(z)) =S(r,f), and the hyper order of a meromor-phic functionf is defined by

ρ(f) =lim sup

r→∞

log logT(r,f)

logr .

Recently, the topic of a diﬀerence equation and a diﬀerence product in the complex plane

Chas attracted many mathematicians, a number of papers have focused on value

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bution and uniqueness of diﬀerences and diﬀerences operator analogues of Nevanlinna theory (including [–]).

For a transcendental meromorphic functionf of ﬁnite order, a nonzero complex con-stant candα(z)∈S(f), Liuet al.[], Chen et al.[], Luo and Lin [] studied the ze-ros distributions of diﬀerence polynomials of meromorphic functions and obtained: If

n≥, then f(z)n_f₍_z₊_c_{) –}_α₍_z_{) has inﬁnitely many zeros [, Theorem .]; if} _n_>_m_, then P(f)f(z+c) –a(z) has inﬁnitely many zeros (see [, Theorem ]), where P(z) =

anzn+an–zn–+· · ·+az+ais a nonzero polynomial, wherea,a, . . . ,an(= ) are com-plex constants andmis the number of the distinct zeros ofP(z).

In , Liuet al.[] studied the zeros distribution of difference-differential polynomi-als, which are the derivatives of difference products of entire functions, and obtained the result as follows.

Theorem .(see [, Theorem .]) Let f be a transcendental entire function of finite order.If n≥k+ ,then the difference-differential polynomial[f(z)n_f₍_z₊_c_)](k)_–_α₍_z₎_has infinitely many zeros.

In the same year, Chen and Chen [] studied the zeros of diﬀerence polynomialsfn(fm– )d_j_=f(z+cj)sj, wherecj∈C\ {}(j= , . . . ,d) are distinct constants,n,d,sj(j= , . . . ,d)∈

N+, and obtained the following theorem.

Theorem .(see [, Theorem .]) Let f be a transcendental entire function of ﬁnite order.If n≥,then fn₍_fm_{– )}d

j=f(z+cj)sj–α(z)has inﬁnitely many zeros.

Let

F(z) =P(f) d

j=

f(z+cj)sj, ()

F(z) =P(f) d

j=

f(z+cj) –f(z)

sj

. ()

We investigate the value distribution of diﬀerence polynomials of a more general form

F(z)(k)–α(z) =

P(f)

d

j=

f(z+cj)sj

(k)

–α(z),

F(z)(k)–α(z) =

P(f)

d

j=

f(z+cj) –f(z)

sj

(k)

–α(z),

whereP(f) is a nonzero polynomial of degreen,mis the number of the distinct zeros of

P(z),cj∈C\ {}(j= , . . . ,d) are distinct constants,n,d,sj(j= , . . . ,d)∈N+, andα(z) is a small function off. Setλ=s+s+· · ·+sd, we obtain the following results.

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Theorem . Let f be a transcendental meromorphic(resp.entire)function ofρ(f) < , and let F(z)be stated as in().If cj∈C\ {}(j= , . . . ,d)are distinct constants,n,d,sj(j= , . . . ,d)∈N+satisfy n> (m+ d)(k+ ) +λ+d+  (resp.n> (m+ d)(k+ )),then(F(z))(k)–

α(z)has inﬁnitely many zeros,provided that f(z+cj)=f(z)for j= , , . . . ,d.

Letf(z) andg(z) be two nonconstant meromorphic functions for somea∈C∪ {∞}. If the zeros off(z) –aandg(z) –a(ifa=∞, zeros off(z) –aandg(z) –aare the poles of

f(z) andg(z) respectively) coincide in locations and multiplicities, we say thatf(z) andg(z) share the value aCM(counting multiplicities), and if they coincide in locations only, we say thatf(z) andg(z) share aIM(ignoring multiplicities).

For the uniqueness of diﬀerence of meromorphic functions, some results have been ob-tained (see [, , –]). Here, we only state some newest theorems as follows.

Theorem .[, Theorem ] Let f and g be transcendental entire functions of ﬁnite order,

let c be a nonzero complex constant,let P(z)be stated as in Theorem.,and let n> +  be an integer,where=m+ m,mis the number of simple zeros of P(z),and mis the number of multiple zeros of P(z).If P(f)f(z+c)and P(g)g(z+c)shareCM,then one of the following results holds:

(i) f≡tgfor a constanttsuch thattl= ,wherel=GCD{λ,λ, . . . ,λn}and

λi=

i+ , ai= ,

n+ , ai= ,

i= , , , . . . ,n.

(ii) f andgsatisfy the algebraic equationR(f,g)≡,where R(ω,ω) =P(ω)ω(z+c) –P(ω)ω(z+c);

(iii) f(z) =eα(z)_,_g₍_z_{) =}_eβ(z)_,_where_α₍_z₎_and_β₍_z₎_{are two polynomials}_,_b_{is a constant} satisfyingα+β≡banda

ne(n+)b= .

In this paper, we investigate the uniqueness problem of diﬀerence polynomials of entire functions

F*(z) =fn fm–  d

j=

f(z+cj)

sj

and G*(z) =gn gm–  d

j=

g(z+cj)

sj

and obtain the following results.

Theorem . Let f,g be transcendental entire functions ofρ(f),ρ(g) < .If(F*(z))(k)and

(G*₍_z₎₎(k)_share__{CM and n}_>_max_{_(_k₊_d_{+ ) +}_m_–_λ_{, }_d_{+  +}_λ_,_k_{+ }_}_,_{where c}

j∈C,

n,m,k,d,sj(j= , , . . . ,d)∈N+,then f ≡tg for a constant t such that tκ= ,whereκ=

GCD{m,n+λ}.

Theorem . Under the assumptions of Theorem.,if(F*(z))(k)and(G*(z))(k)shareIM and n>max{(k+d) + m+  –λ,k+ },then the conclusions of Theorem.hold.

2 Some lemmas

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Lemma .[] Let f be a nonconstant meromorphic function,and let P(f) =a+af +

af₊_{· · ·}₊_a

nfn,where a,a,a, . . . ,anare constants and an= .Then

T r,P(f)=nT(r,f) +S(r,f).

Lemma .[, Theorem .] Let f be a transcendental meromorphic function ofρ(f) < ,

ς< ,and letεbe an enough small number.Then

m

r,f(z+c)

f(z)

=o

T(r,f)

r–ς–ε

=S(r,f)

for all r outside of a set of ﬁnite logarithmic measure.

Lemma .[, Lemma .] Let T: [, +∞)→[, +∞)be a non-decreasing continuous function,and let s∈(, +∞).If the hyper order of T is strictly less that one,that is,

lim sup

r→∞

log logT(r)

logr < ,

andδ∈(,  –ς),then

T(r+s) =T(r) +o

T(r)

rδ

for all r runs to inﬁnity outside of a set of ﬁnite logarithmic measure.

From Lemma ., we can get the following lemma easily.

Lemma . Let f(z)be a transcendental meromorphic function of hyper orderρ(f) < , and let c be a nonzero complex constant.Then we have

T r,f(z+c)=T r,f(z)+S(r,f), N r,f(z+c)=N r,f(z)+S(r,f),

and

N

r, 

f(z+c)

=N

r,

f

+S(r,f).

Lemma . Let f be a transcendental meromorphic function ofρ(f) < ,let F(z)be stated as in().Then we have

(n–λ)T(r,f) +S(r,f)≤T r,F(z)≤(n+λ)T(r,f) +S(r,f). ()

If f is a transcendental entire function ofρ(f) < ,we have

T r,F(z)=T r,P(f)fλ₊_S₍_r_,_f_{) = (}_n₊_λ₎_T₍_r_,_f_{) +}_S₍_r_,_f_), _()

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Proof Iff is a transcendental entire function ofρ(f) < , from Lemma .-Lemma ., we

have

T r,F(z)=m r,F(z)≤m r,P(f)fλ(z)+m

r,

d

j=f(z+cj)sj

fλ₍_z₎

≤m r,P(f)fλ(z)+S(r,f) =T r,P(f)fλ(z)+S(r,f) = (n+λ)T(r,f) +S(r,f).

On the other hand, from Lemma ., we have

(n+λ)T(r,f) =T r,P(f)fλ₍_z₎₊_S₍_r_,_f_{) =}_m _r_,_P₍_f₎_fλ₍_z₎₊_S₍_r_,_f₎

≤m r,F(z)+m

r, f λ₍_z₎

d

j=f(z+cj)sj

+S(r,f)

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

Thus, we can get ().

Iff is a meromorphic function ofρ(f) < , from Lemma . and Lemma ., we have

T

r,P(f) d

i=

f(z+cj)sj

≤T r,P(f)+T

r, d

i=

f(z+cj)sj

≤(n+λ)T(r,f) +S(r,f).

On the other hand, from Lemma . and Lemma ., we have

(n+λ)T(r,f) =T r,P(f)fλ₊_S₍_r_,_f_{) =}_m _r_,_P₍_f₎_fλ₊_N _r_,_P₍_f₎_fλ₊_S₍_r_,_f₎

≤m

r,F(z) f λ₍_z₎

d

j=f(z+cj)sj

+N

r,F(z) f λ₍_z₎

d

j=f(z+cj)sj

+S(r,f)

≤T r,F(z)+ λT(r,f) +S(r,f).

Thus, we can get ().

Using the same method as in Lemma ., we can get the following lemma easily.

Lemma . Let f be a transcendental meromorphic function ofρ(f) < ,and let F(z)be stated as in().Then we have

(n–λ)T(r,f) +S(r,f)≤T r,F(z)≤(n+ λ)T(r,f) +S(r,f).

If f is a transcendental entire function ofρ(f) < ,we have

T r,F(z)=T

r,P(f) d

j=

f(z+cj) –f(z)

sj

≥nT(r,f) +S(r,f).

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off, we ignore the multiplicities (see [, ]) andNk(r,_f_–_a) =N(r,_f_–_a) +N(r,_f_–_a) +· · ·+

Nk(r,_f_–_a).

Deﬁnition .[] Whenf andgshare IM, we denote byNL(r,_f_–) the counting func-tion of the -points off whose multiplicities are greater than -points ofg, where each zero is counted only once; similarly, we haveNL(r,_g_– ). Letzbe a zero off –  of mul-tiplicitypand a zero ofg–  of multiplicityq. We also denote byN(r,_f_– ) the counting

function of those -points off wherep=q= .

Lemma .[] and [, Lemma .] Let f be a nonconstant meromorphic function,and let p,k be positive integers.Then

T r,f(k)≤T(r,f) +kN(r,f) +S(r,f),

Np

r, 

f(k)

≤T r,f(k)–T(r,f) +Np+k

r,

f

+S(r,f),

Np

r, 

f(k)

≤kN(r,f) +Np+k

r,

f

+S(r,f).

Lemma .[] Let f and g be two meromorphic functions.If f and g shareCM,then one of the following three cases holds:

(i)

T(r,f) +T(r,g)≤N(r,f) + N(r,g) + N

r,

f

+ N

r,

g

+S(r,f) +S(r,g);

(ii) f≡g; (iii) f·g= .

Lemma .[] Let f and g be two meromorphic functions.If f and g shareIM,then one of the following cases must occur:

(i)

T(r,f) +T(r,g)≤

N(r,f) +N

r,

f

+N(r,g) +N

r,

g

+ NL

r, 

f– 

+ NL

r, 

g– 

+S(r,f) +S(r,g);

(ii) f =(b+)_bgg₊₍+(_aa_––_bb₎–),wherea(= ),bare two constants.

Lemma . [, Theorem .] Let a(z), . . . ,an(z),b(z) be polynomials such that

a(z)an(z)≡,deg(

degaj=daj) =d,where d=max≤j≤n{degaj}.If f(z)is a transcendental

meromorphic solution of

a(z)f(z) +a(z)f(z+cj) +· · ·+an(z)f(z+cn) =b(z),

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cμ, dμ are positive integers, iι,jι ∈ N (ι = , , . . . ,k – ), I ={i,i, . . . ,ik–} and J =

{j,j, . . . ,jk–}satisfy

i+ i+· · ·+ (k– )ik–=k– , j+ j+· · ·+ (k– )jk–=k– .

From (), we have (U_+A_U)eA_{– (}_U

+AU)eA has no zeros. IfU+AU= , then U_+A_U= . IfAis transcendental entire, we have

m r,A_=m

r,U

 U

=S r,U_=o T r,U_. ()

SinceA,Aare entire, we have

T r,A_i=mr,A_i, T r, A_i(k)≤T r,A_i+S(r,Ai), k∈N,i= , . ()

Thus, from (), () and the deﬁnition ofU, we can getT(r,A_)≤o(T(r,A_)), a contradic-tion withA_is transcendental entire. Hence,Ais a polynomial. From Lemma ., we can getρ(f)≥, which is a contradiction. IfU+AU= , similar to the above argument, we

can get a contradiction.

Therefore, we can get thatf ≡g.

3 Proofs of Theorems 1.3 and 1.4 3.1 The Proof of Theorem 1.3

From (), by Lemma . and Lemma ., we have thatF(z) is not constant andS(r,F(k)_{) =} S(r,F) =S(r,f). Next, we consider two cases.

Case . Iff is a transcendental meromorphic function ofρ(f) < , we ﬁrst suppose that

(P(f)d_j_=f(z+cj)sj)(k)=a(z) has ﬁnitely solutions. By the second fundamental theorem for three small functions (see [, Theorem .]) and Lemma ., we have

T r,F(k)≤N r,F(k)+N

r, 

F(k)

+N

r, 

F(k)_–_a₍_z₎

+S r,F(k)

≤N(r,f) + d

j=

N r,f(z+cj)

+N

r, 

F(k)

+S r,F(k)

≤(d+ )T(r,f) +T r,F(k)–T(r,F) +Nk+

r,

F

+S r,F(k).

By Lemma ., Lemma . and Lemma ., we obtain

(n–λ)T(r,f) +S(r,f)≤T(r,F)≤(d+ )T(r,f) +Nk+

r, 

F

+S(r,f)

≤(d+ )T(r,f) +m(k+ )N

r,

f

+ d

j= N

r, 

f(z+cj)

+S(r,f)

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Fromf is transcendental andn>m(k+ ) + d+  +λ, we can get a contradiction to (). ThenF(z)(k)–a(z) has inﬁnitely many zeros.

Case . Iff is a transcendental entire function ofρ(f) < . Suppose that (P(f) d

j=f(z+ cj)sj)(k)=a(z) has ﬁnitely solutions. By using the same argument as in Case  and (), we have

(n+λ)T(r,f) +S(r,f)≤m(k+ ) +dT(r,f) +S(r,f),

which is a contradiction withn>m(k+ ) +d–λ. Thus, we complete the proof of Theorem ..

3.2 The Proof of Theorem 1.4 We consider two cases as follows.

Case . Suppose thatfis a transcendental meromorphic function ofρ(f) < . Similar to

the proof of Case  in Theorem ., and using Lemma ., we have

(n–λ)T(r,f) +S(r,f)≤T(r,F)≤(d+ )N(r,f) +Nk+

r, 

F

+S(r,f)

≤(d+ )T(r,f) +m(k+ )N

r,

f

+ (k+ ) d

j= N

r, 

f(z+cj) –f(z)

+S(r,f)

≤(m+ d)(k+ ) +d+ T(r,f) +S(r,f).

Sincen> (m+ d)(k+ ) +d+  +λandf is transcendental, we can get a contradiction. Case . Suppose thatf is a transcendental entire function ofρ(f) < . Similar to the

proof of Case  in Theorem ., by Lemma ., we can get

nT(r,f)≤(m+ d)(k+ )T(r,f) +S(r,f),

which is a contradiction withn> (m+ d)(k+ ) andf is transcendental. Thus, from Cases  and , we prove Theorem ..

4 Proofs of Theorems 1.6 and 1.7 4.1 The proof of Theorem 1.6

LetH(z) = (F*₍_z₎₎(k)_and_G₍_z_{) = (}_G*₍_z₎₎(k)_{. From the assumptions of Theorem ., we have}

thatH(z),G(z) share CM. By Lemma ., we have

T r,H(z)≤T

r,fn fm–  d

j=

f(z+cj)

sj

+S

r,fn fm–  d

j=

f(z+cj)

sj

.

By Lemma . and Lemma ., we can getS(r,H) =S(r,f). Similarly, we haveS(r,G) =

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Sincef,gare transcendental andn> (k+d+ ) +m–λ, we can get a contradiction. Case . IfH(z)≡G(z), then we have

fn fm–  d

j=

f(z+cj)

sj

=gn gm–  d

j=

g(z+cj)

sj

+Q(z), ()

whereQ(z) is a polynomial of degree at mostk– . IfQ(z)≡, by the second fundamental theorem and Lemma ., we have

(n+m+λ)T(r,f) =T

r,f

n₍_fm_{– )}d

j=(f(z+cj))sj

Q(z)

+S(r,f)

≤N

r,f

n₍_fm_{– )}d

j=(f(z+cj))sj

Q(z)

+N

r, Q(z)

fn₍_fm_{– )}d

j=(f(z+cj))sj

+N

r, Q(z)

gn₍_gm_{– )}d

j=(g(z+cj))sj

+S(r,f)

≤(m+d+ )T(r,f) +T(r,g)+S(r,f) +S(r,g). ()

Similarly, we have

(n+m+λ)T(r,g)≤(m+d+ )T(r,f) +T(r,g)+S(r,f) +S(r,g). () From (), () andn> (k+d+ ) +m–λ> d+m+  –λ, we can get a contradiction. Thus,Q(z)≡, that is,

fn fm–  d

j=

f(z+cj)

sj

≡gn gm–  d

j=

g(z+cj)

sj

. ()

Seth=f_g. Ifhis a constant. Substitutingf =ghinto (), we can get

d

j=

g(z+cj)sj

gn+m hn+m+λ– +gn hn+λ– ≡. ()

Sincegis a transcendental entire function, we haved_j_=g(z+cj)sj≡. Then, from (), we have

gm hn+m+λ– + hn+λ– ≡. ()

Suppose thathn+λ_{= , by Lemma . and (), we have}_T₍_r_,_g_{) =}_S₍_r_,_g_{), which is} contra-diction with a transcendental functiong. Thenhn+λ_{= . Similar to this discussion, we can} get thathn+m+λ_{= . If}_h_{is not a constant, from () we have}

g(z)m= h(z) nd

j=(h(z+cj))sj– 

h(z)n+md

j=(h(z+cj))sj– 

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If  is a Picard value ofh(z)n+md

Then from () and by the second fundamental theorem and Lemma ., we have

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4.2 The proof of Theorem 1.7 We consider three cases as follows.

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From () and (), we have

(n+λ– k– d–m– )T(r,f) +T(r,g)≤S(r,f) +S(r,g), which is a contradiction withn> (k+d) + m+  –λ.

If a–b–  = , then by () we knowH= ((b+ )G)/(bG+ ). Sincef, g are entire functions, we get that –_b is a Picard exceptional value ofG(z). By the second fundamental theorem, we have

(n+m+λ)T(r,g)≤T(r,G) +Nk

r, 

gn₍_gm_{– )}d

j=(g(z+cj))sj

–N

r, 

G

+S(r,g)

≤Nk

r, 

gn₍_gm_{– )}d

j=(g(z+cj))sj

+N

r, 

G+_b

+S(r,g)

≤(k+m+d)T(r,g) +S(r,g), which is a contradiction withn> (k+d) + m+  –λ.

Subcase ..b= –. Then () becomesH=a/(a+  –G).

Ifa+ = , thena+  is a Picard exceptional value ofG. Similar to the discussion in Subcase ., we can deduce a contradiction again.

Ifa+  = , thenHG≡. From Lemma ., we can get thatf ≡g. Subcase ..b= . Then () becomesH= (G+a– )/a.

Ifa– = , thenN(r,_G₊_a_–) =N(r,_H). Similar to the discussion in Subcase ., we can deduce a contradiction again.

Ifa–  = , thenH≡G. Using the same argument as in the proof of Case  in Theo-rem ., we can get thatf,gsatisfyf ≡tgfor a constanttsuch thattm_{=  and}_tn+λ_{= .}

Thus, we complete the proof of Theorem ..

Competing interests

The author declares that they have no competing interests.

Author’s contributions

HYX completed the main part of this article, HYX corrected the main theorems. The author read and approved the ﬁnal manuscript.

Acknowledgements

The author thanks the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (61202313), the Natural Science Foundation of Jiang-Xi Province in China (Grant No. 2010GQS0119 and No. 20122BAB201016).

Received: 5 January 2013 Accepted: 14 March 2013 Published: 3 April 2013

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doi:10.1186/1687-1847-2013-90