R E S E A R C H
Open Access
Results on the growth of meromorphic
solutions of some linear difference equations
with meromorphic coefficients
Zhang Li Yuan and Qiu Ling
**Correspondence:
qiuling1978@bjut.edu.cn College of Applied Science, Beijing University of Technology, Beijing, 100124, China
Abstract
In this paper, we investigate the growth of meromorphic solutions of some linear difference equations. We obtain some new results on the growth of meromorphic solutions when most coefficients in such equations are meromorphic functions, which are supplements of previous results due to Li and Chen (Adv. Differ. Equ. 2012:203, 2012) and Liu and Mao (Adv. Differ. Equ. 2013:133, 2013).
MSC: 30D35; 39A10
Keywords: complex difference equation; meromorphic coefficients; growth
1 Introduction and main results
In this article, a meromorphic function always means meromorphic in the whole complex planeC, andcalways means a nonzero constant. We adopt the standard notations of the Nevanlinna value distribution theory of meromorphic functions such asT(r,f),m(r,f) andN(r,f) as explained in [–]. In addition, we will use notationsρ(f) to denote the order of growth of a meromorphic functionf(z),λ(f) to denote the exponents of convergence of the zero sequence of a meromorphic functionf(z), λ(
f) to denote the exponents of
convergence of the pole sequence of a meromorphic functionf(z), and we define them as follows:
ρ(f) =lim sup
r→∞
logT(r,f)
logr ,
λ(f) =lim inf
r→∞
logN(r,f)
logr ,
λ
f
=lim inf
r→∞
logN(r,f)
logr .
Recently, meromorphic solutions of complex difference equations have become a sub-ject of great interest from the viewpoint of Nevanlinna theory due to the apparent role of the existence of such solutions of finite order for the integrability of discrete difference equations.
About the growth of meromorphic solutions of some linear difference equations, some results can be found in [–]. Laine and Yang [] considered the entire functions coef-ficients case and got the following.
Theorem A[] Let Aj(z) (≡) (j= , , . . . ,n)be entire functions of finite order such that
among those coefficients having the maximal orderρ:=max≤j≤nρ(Aj),exactly one has its
type strictly greater than the others.Then,for any meromorphic solution f(z)to
An(z)f(z+n) +· · ·+A(z)f(z+ ) +A(z)f(z) = ,
we haveρ(f)≥ρ+ .
Chiang and Feng [, ] improved Theorem A as follows.
Theorem B[, ] Let Aj(z) (≡) (j= , , . . . ,n)be entire functions such that there exists
an integer l, ≤l≤n,such that
ρ(Al) > max
≤j≤n,j=lρ(Aj).
Suppose that f(z)is a meromorphic solution to
An(z)f(z+n) +· · ·+A(z)f(z+ ) +A(z)f(z) = ,
then we haveρ(f)≥ρ(Al) + .
Recently in [], Peng and Chen investigated the order and the hyper-order of solutions of some second-order linear differential equations and proved the following results.
Theorem C[] Suppose that Aj(z) (≡) (j= , )are entire functions andρ(Aj) < .Let
α,αbe two distinct complex numbers such that αα= ( suppose that|α| ≤ |α|).If
argα=πorα< –,then every solution f(z)≡of the equation
f+e–zf+Aeαz+Aeαz
f =
has infinite order andρ(f) = .
Moreover, Xu and Zhang [] extended the above result from entire coefficients to mero-morphic coefficients.
It is well known that f(z) =f(z+c) –f(z) is regarded as the difference counterpart off. Thus a natural question is: Can we change the above second-order linear differen-tial equation to the linear difference equation? What conditions will guarantee that every meromorphic solution will have infinite order when most coefficients in such equations are meromorphic functions?
Li and Chen [] considered the following difference equation and obtained the follow-ing theorem.
Theorem D[] Let k be a positive integer, p be a nonzero real number and f(z)be a nonconstant meromorphic solution of the difference equation
An(z)f(z+n) +· · ·+A(z)f(z+ ) +
where Aj(z),B(z),B(z) (≡) (j= , , . . . ,n)are all entire functions andmax{ρ(Aj),ρ(B),
ρ(B)}=σ<k,then we haveρ(f)≥k+ .
The main purpose of this paper is to investigate the growth of meromorphic solutions of certain linear difference equations with meromorphic coefficients. The remainder of the paper studies the properties of meromorphic solutions of a nonhomogeneous linear difference equation. In fact, we prove the following results, in which there are still some coefficients dominating in some angles.
Theorem . Let k be a positive integer.Suppose that Aj(z),B(z) (≡) (j= , , . . . ,n)are
all entire functions andmax{ρ(Aj),ρ(B) : ≤j≤n}=α<k.Letα,βbe two distinct
complex numbers such thatαβ= (suppose that|α| ≤ |β|).Letαbe a strictly negative
real constant.Ifargα=π orα<α, then every meromorphic solution f(z)≡of the
equation
An(z)f(z+n) +· · ·+A(z)f(z+ ) +A(z)eαz k
f(z+ )
+A(z)eαzk+B(z)eβzkf(z) = (.)
satisfiesρ(f)≥k+ .
Theorem . Suppose that Aj(z),B(z) (≡) (j= , , . . . ,n)are all meromorphic functions
andmax{ρ(Aj),ρ(B) : ≤j≤n}=α< .Letα,βbe two distinct complex numbers such
thatαβ= (suppose that|α| ≤ |β|).Letαbe a strictly negative real constant.Ifargα= πorα<α,then every meromorphic solution f(z)≡of the equation
An(z)f(z+n) +· · ·+A(z)f(z+ ) +A(z)eαzf(z+ )
+A(z)eαz+B(z)eβz
f(z) = (.)
satisfiesρ(f)≥.
Theorem . Under the assumption for the coefficients of(.)in Theorem.,if f(z)is a finite order meromorphic solution to(.),thenλ(f–z) =ρ(f).What is more,either k+ ≤ ρ(f)≤max{λ(f),λ(f)}+ orρ(f) =k+ >max{λ(f),λ(f)}+ .
Theorem . Under the assumption for the coefficients of(.)in Theorem.,if f(z)is a finite order meromorphic solution to(.),thenλ(f –z) =ρ(f).What is more,either≤ ρ(f)≤max{λ(f),λ(f)}+ orρ(f) = >max{λ(f),λ(f)}+ .
Liu and Mao [] considered the meromorphic solutions of the difference equation
ak(z)f(z+k) +· · ·+a(z)f(z+ ) +a(z)f(z) = , (.)
one of their results can be stated as follows.
Theorem E [] Let aj(z) =Aj(z)ePj(z) (j= , , . . . ,k),where Pj(z) =αjnzn+· · ·+αjare
, , . . . ,k)are distinct complex numbers,then every meromorphic solution f (≡)of Eq. (.)satisfiesρ(f)≥max≤j≤k{ρ(aj)}+ .
In this paper, we extend and improve the above result from entire coefficients to mero-morphic coefficients in the case where the polynomialsPj(z) are of degree .
Theorem . Let aj(z) =Aj(z)eαjz(j= , , . . . ,n),αjare distinct complex constants,suppose
that Aj(z) (≡)are meromorphic functions andρ(Aj) =α< ,then every meromorphic
solution f (≡)of the equation
an(z)f(z+n) +· · ·+a(z)f(z+ ) +a(z)f(z) = (.)
satisfiesρ(f)≥.
Theorem . Let aj(z) =Aj(z)eαjz+Dj(z) (j= , , . . . ,n),αjare distinct complex constants,
suppose that Aj(z) (≡),Dj(z) (≡)are meromorphic functions andρ(Aj) =α< ,ρ(Dj) =
β< ,then every meromorphic solution f (≡)of Eq. (.)satisfiesρ(f)≥.
Next we consider the properties of meromorphic solutions of the nonhomogeneous lin-ear difference equation corresponding to (.)
an(z)f(z+n) +· · ·+a(z)f(z+ ) +a(z)f(z) =F(z), (.) whereF(z) (≡) is a meromorphic function.
Theorem . Let aj(z) (j= , , . . . ,n)satisfy the hypothesis of Theorem.or Theorem.,
and let F(z)be a meromorphic function ofρ(F) < ,then at most one meromorphic solution fof Eq. (.)satisfies≤ρ(f)≤andmax{λ(f),λ(f)}=ρ(f),the other solutions f satisfy ρ(f)≥.
2 Some lemmas
In this section, we present some lemmas which will be needed in the sequel.
Lemma .[] Let f(z)be a meromorphic function of finite orderρ, be a positive con-stant,ηandηbe two distinct nonzero complex constants.Then
m
r,f(z+η)
f(z+η)
=Orρ–+ ,
and there exists a subset E⊂(,∞)of finite logarithmic measure such that,for all z
satis-fying|z|=r∈/[, ]∪E,and as r→ ∞,
exp–rρ–+ ≤f(z+η)
f(z+η)
≤exprρ–+.
Lemma .[] Let f(z)be a meromorphic function of finite orderρ,then,for any given
> ,there exists a set E⊂(,∞)of finite logarithmic measure such that,for all z satisfying
|z|=r∈/[, ]∪E,and as r→ ∞,
Lemma .[] Suppose that P(z) = (α+iβ)zn+· · · (α,βare real numbers,|α|+|β| = )
is a polynomial with degree n≥,A(z) (≡)is an entire function withρ(A) <n.Set g(z) =
A(z)eP(z),z=reiθ,δ(P,θ) =αcosnθ–βsinnθ.Then,for any given > ,there exists a set
E⊂[, π)that has linear measure zero such that for anyθ∈[, π)\(E∪E),there is
R> such that for|z|=r>R,we have
(i) ifδ(P,θ) > ,then
exp( – )δ(P,θ)rn<greiθ<exp( + )δ(P,θ)rn;
(ii) ifδ(P,θ) < ,then
exp( + )δ(P,θ)rn<greiθ<exp( – )δ(P,θ)rn,
where E={θ∈[, π) :δ(P,θ) = }is a finite set.
Lemma . applies in Theorem . whereA(z) (≡) is an entire function.
Lemma .[] Suppose that n≥is a positive integer.Let Pj(z) =ajnzn+· · · (j= , )
be nonconstant polynomials, where ajq (q= , . . . ,n)are distinct complex numbers and
anan= .Set z=reiθ,ajn=|ajn|eiθj,θj∈[–π,π),δ(Pj,θ) =|ajn|cos(θj+nθ),then there is
a set E⊂[–πn,πn)that has linear measure zero.Ifθ=θ,then there exists a rayargz=θ, θ∈[–πn,πn)\(E∪E)such that
δ(P,θ) > , δ(P,θ) <
or
δ(P,θ) < , δ(P,θ) > ,
where E={θ∈[–πn,πn) :δ(Pj,θ) = }is a finite set,which has linear measure zero.
In Lemma ., ifθ ∈[–π n,
π
n)\(E∪E) is replaced byθ ∈[
π n,
π
n)\(E∪E), then we
have the same result.
Lemma .[] Consider g(z) =A(z)eaz,where A(z) (≡)is a meromorphic function with
ρ(A) =α< ,a is a complex constant,a=|a|eiϕ(ϕ∈[, π)).Set E
={θ∈[, π) :cos(ϕ+ θ) = },then Eis a finite set.Then,for any given ( < < –α),there exists a set E⊂ [, π)that has linear measure zero,if z=reiθ,θ∈[, π)\(E
∪E),then we have(when r
is sufficiently large): (i) ifδ(az,θ) > ,then
exp( – )δ(az,θ)r≤greiθ≤exp( + )δ(az,θ)r; (ii) ifδ(az,θ) < ,then
exp( + )δ(az,θ)r≤greiθ≤exp( – )δ(az,θ)r,
Lemma . applies in Theorem . whereA(z) (≡) is a meromorphic function.
for allroutside of a possible exceptional setEwith finite logarithmic measure. Applying Lemma ., we have
for allroutside of a possible exceptional setEwith finite linear measure. Therefore from (.) we can get that
Since θ ∈ [–πk,πk)\(E ∪E ∪ E ∪E), we know that coskθ > , then |eαz k
| =
e–|α|rkcoskθ< . Therefore, by (.) we obtain
A(z)eαzk≤exprα+ . (.)
Substituting (.), (.), (.), (.) and (.) into (.), we obtain
Mexp
( + )δαzk,θ
rk≤nexprα+ +rρ(f)–+ . (.)
Byδ(αzk,θ) > ,α+ <kandρ(f) – + <k, we know that (.) is a contradiction. Case .α<α, which isθ=π.
Subcase .. Assume thatθ=ϕ, thenϕ=π. By Lemma ., for the above , there is a rayargz=θ such thatθ∈[–πk,πk)\(E∪E∪E∪E) (whereEandEare defined as in Lemma .,E∪E∪E∪Eis of linear measure zero) satisfyingδ(βzk,θ) > . Since
coskθ> , we haveδ(αzk,θ) =|α|cos(θ+kθ) = –|α|coskθ< . For a sufficiently larger, by Lemma ., we get
A(z)eαz k
<exp( – )δαzk,θ
rk< , (.)
B(z)eβzk>exp( – )δβ zk,θ
rk. (.)
By (.) and (.), we get
A(z)eαzk+B(z)eβzk≥B(z)eβzk–A(z)eαzk
>exp( – )δβzk,θ
rk– . (.)
Using the same reasoning as in Subcase ., we can get a contradiction.
Subcase .. Assume thatθ=ϕ, thenθ=ϕ=π. By Lemma ., for the above , there is a rayargz=θ such thatθ∈[π
k,
π
k)\(E∪E∪E∪E) (whereEandEare defined
as in Lemma .,E∪E∪E∪Eis of linear measure zero), thencoskθ< ,δ(αzk,θ) =
|α|cos(θ+kθ) = –|α|coskθ> ,δ(βzk,θ) =|β|cos(ϕ+kθ) = –|β|coskθ> . Since|α| ≤ |β|,α=β, andθ=ϕ, then|α|<|β|, thusδ(βzk,θ) >δ(αzk,θ) > .
For a sufficiently larger, we get (.), (.) and (.) hold.
Using the same reasoning as in Subcase ., we can get a contradiction. Thus we have ρ(f)≥k+ .
3.2 The proof of Theorem 1.2
SinceAj(z),B(z) (≡) (j= , , . . . ,n) are meromorphic functions andmax{ρ(Aj),ρ(B) :
≤j≤n}=α < , then when δ(αz,θ) > ,β(az,θ) < for a sufficiently large r, by Lemma ., we have
A(z)eαz≥exp( – )δ(α z,θ)r
, (.)
B(z)eβz≤exp
( – )δ(βz,θ)r
< . (.)
3.3 The proof of Theorems 1.3 and 1.4
Now, for any given > , applying Lemma . and Theorem ., we can deduce that
m torization of a meromorphic function, we writef(z) as follows:
where
tion to our assumption. This completes our proof.
The proof of Theorem . is similar to that of Theorem ..
3.4 The proof of Theorem 1.5
By (.), (.) and (.), we get forz=reiθand sufficiently larger∈/[, ]∪E,
any given > , there exists a subsetE⊂(,∞) of finite logarithmic measure such that, for allzsatisfying|z|=r∈/[, ]∪E, and asr→ ∞, we have
Then, using a similar argument to that of Theorem . and Theorem . and only replacing (.) (or (.)) by (.) (or (.)), we can prove Theorem ..
3.6 The proof of Theorem 1.7
Letf(z)≡ be a meromorphic solution of (.). Suppose thatρ(f) < , then by Lemma ., we obtain ρ(F) =ρ( kj=aj(z)f(z+j)) = . This contradictsρ(F) < , therefore we have
ρ(f)≥.
linear difference equation corresponding to (.) and ρ(f–f) < . By Theorem . or Theorem ., we get a contradiction. So Eq. (.) has at most one meromorphic solution
fsatisfying ≤ρ(f)≤. Next we prove max{λ(f),λ(f
)}=ρ(f) in the caseρ(f) = . Suppose that max{λ(f), λ(f
)}<ρ(f), then by the Weierstrass factorization, we obtain
f(z) =
P(z)
P(z)e
Q(z), (.)
whereP(z),P(z) are entire functions such thatρ(P) =λ(P) =λ(f),ρ(P) =λ(P) =λ(f ) andQ(z) is a polynomial of degree .
In the caseaj(z) =Aj(z)eαjz. Substituting (.) into (.), we get n
j=
Aj(z)
P(z+j)
P(z+j)e
αjz+Q(z+j)=F(z). (.)
Sinceαjare distinct complex numbers, by Lemma ., we obtain that the order of the
left-hand side of (.) is . This contradictsρ(F) < .
Foraj(z) =Aj(z)eαjz+Dj(z), by using an argument similar to the above, we also obtain a
contradiction.
It is obvious thatmax{λ(f),λ(f
)}=ρ(f) provided that <ρ(f) < . Therefore we have
max{λ(f),λ(f
)}=ρ(f).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZLY performed and drafted manuscript. All authors read and approved the final manuscript.
Acknowledgements
The author thanks the referee for his/her valuable suggestions to improve the present article. The research was supported by the Beijing Natural Science Foundation (No. 1132013) and the Foundation of Beijing University of Technology (No. 006000514313002).
Received: 9 July 2014 Accepted: 14 November 2014 Published:03 Dec 2014
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Cite this article as:Yuan and Ling:Results on the growth of meromorphic solutions of some linear difference