R E S E A R C H
Open Access
Growth of solutions of some kinds of
linear difference equations
Shun-Zhou Wu and Xiu-Min Zheng
**Correspondence:
[email protected] Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, P.R. China
Abstract
In this paper, we investigate the growth and value distribution of solutions of some kinds of linear difference equations, where there may be more than one coefficient having the same maximal order and the same maximal type.
MSC: 30D35; 39A22
Keywords: linear difference equation; meromorphic solution; growth; exponent of convergence
1 Introduction
Throughout this paper, we use the standard notations and basic results of Nevanlinna’s value distribution theory (see [–]). In addition, we use the notationsσ(f),σ(f) and λ(f–a) to denote, respectively, the order, the hyperorder, and the exponent of convergence of the sequence ofa-points of a meromorphic functionf(z) in the complex plane, where
a∈C∪ {∞}. Furthermore, we can get the definition ofλ(f –ϕ), whenais replaced by a meromorphic functionϕ(z).
Recently, with the research and further development of difference analogs of Nevan-linna’s theory, it has been applied more and more widely in the difference field. By this important tool, many scholars investigated the linear difference equation
Ak(z)f(z+ck) +· · ·+A(z)f(z+c) +A(z)f(z) = , (.) wherek∈N+ andcj, j= , . . . ,k are distinct nonzero complex constants, and obtained
many results on the growth and the exponent of convergence of the sequence of zeros of meromorphic solutions of (.). For instance, in [–], the authors considered the case when there is exactly only one coefficient of (.) having the maximal order; in [–], the authors considered the case when there is exactly only one coefficient having the type strictly greater than the others among those having the maximal order; and in [, , ], the authors considered the case when the coefficients of (.) are polynomials.
Further, how about the case when there are more than one coefficient having the same maximal order and the same maximal type?
In , Liu [] considered the growth and the exponent of convergence of the sequence of small function value points of second order linear difference equations and obtained the following theorem.
Theorem A(see []) Let c,c(=c),a be nonzero constants,h(z)be a nonzero
mero-morphic functions withσ(h) < ,B(z)be a nonzero meromorphic function.If B(z)satisfies any one of the following three conditions:
(i) σ(B) > andδ(∞,B) > ; (ii) σ(B) < ;
(iii) B(z) =h(z)ebz,wherebis a nonzero constant,h(z)(≡)is a meromorphic
function withσ(h) < ,
then every meromorphic solution f(z) (≡)of the difference equation
f(z+c) +h(z)eazf(z+c) +B(z)f(z) =
satisfies σ(f)≥max{σ(B), }+ .Further, ifϕ(z) (≡) is a meromorphic function with
σ(ϕ) <max{σ(B), }+ ,thenλ(f –ϕ) =σ(f)≥max{σ(B), }+ .
Liu and Mao [] considered the growth of meromorphic solutions of (.), where the coefficients may have the same order and the same type, and obtained the following the-orem.
Theorem B(see []) Let Aj(z) =hj(z)ePj(z)+Dj(z),j= , , . . . ,k,where Pj(z) =ajnzn+· · ·+
ajare polynomials with degree n(≥),hj(z) (≡),Dj(z)are entire functions with order less
than n.If ajn,j= , , . . . ,k are distinct complex numbers,then every meromorphic solution
f(z) (≡)of(.)satisfiesσ(f)≥max≤j≤k{σ(Aj)}+ .
In this paper, we are concerned with the more general problem than Theorems A and B, and obtain the following results, which extend and improve the previous results.
First, we consider the difference equation (.).
Theorem . Let k,n∈N+, Aj(z) = Bj(z)ePj(z) +Dj(z)eQj(z) +Rj(z), j= , , . . . ,k, where
Pj(z) =ajnzn+· · ·+aj,Qj(z) =bjnzn+· · ·+bj,j= , , . . . ,k are polynomials with degree n
and satisfy|ajn| ≥ |bjn|> ,j= , , . . . ,k,Bj(z),Dj(z),Rj(z),j= , , . . . ,k are meromorphic
functions and satisfymax≤j≤k{σ(Bj),σ(Dj),σ(Rj)}=ω<n,Aj(z) –Rj(z)≡,j= , , . . . ,k.
Let cj,j= , . . . ,k be distinct nonzero complex constants.If there exists an i∈ {, , . . . ,k}
such that for all j(=i),|ain| ≥ |ajn|,and
(i) argain=argajn,orargain=argajn,|ain|>|ajn|
and
(ii) argain=argbjn,orargain=argbjn,|ain|>|bjn|
hold simultaneously,then every meromorphic solution f(z) (≡)of(.)satisfiesσ(f)≥
n+ .Further,ifϕ(z) (≡)is a meromorphic function withσ(ϕ) <n+ ,then for every meromorphic solution f(z) (≡)of(.)withσ(f) < ,we haveλ(f –ϕ) =σ(f)≥n+ .
Remark . In Theorem ., if there exist somej(=i) such thatAj(z) –Rj(z)≡, or some
ofBj(z) (j=i),Dj(z),Rj(z),j= , , . . . ,k, are equal to zero, then the corresponding result
holds by using a similar proof to the one of Theorem ..
Next, we consider difference operators instead of shift operators in (.).
For a nonzero complex constantc, the forward differenceskf(z),k∈N+, are defined (see []) by
It is shown in [] that
kf(z) =
k
j=
Ckj(–)k–jf(z+jc), f(z+kc) =
k
j=
Cjkjf(z), k∈N+.
Then we can obtain the following theorem.
Theorem . Suppose that Aj(z),j= , , . . . ,k,satisfy the hypotheses of Theorem.,and
i is also defined as in Theorem..If i= ,then every meromorphic solution f(z) (≡)of the difference equation
Ak(z)kf(z) +· · ·+A(z)f(z) +A(z)f(z) = (.)
satisfiesσ(f)≥n+ .Further,ifϕ(z) (≡)is a meromorphic function withσ(ϕ) <n+ ,
then for every meromorphic solution f(z) (≡)of(.)withσ(f) < ,we haveλ(f –ϕ) = σ(f)≥n+ .
In the end, we can easily get the following corollary.
Corollary . Let aj∈C,j= , , . . . ,k,such that a=aj,|a| ≥ |aj| ≥,j= , . . . ,k,and
hj(z) (≡),j= , , . . . ,k,be meromorphic functions with order less than n,then every
mero-morphic solution f(z) (≡)of the difference equation
hk(z)eakz
n
kf(z) +hk–(z)eak–z
n
k–f(z) +· · ·+h(z)eaz
n
f(z)
+h(z)eaz
n f(z) =
satisfiesσ(f)≥n+ .Further,ifϕ(z) (≡)is a meromorphic function withσ(ϕ) <n+ ,then for every meromorphic solution f(z) (≡)withσ(f) < ,we haveλ(f –ϕ) =σ(f)≥n+ .
2 Preliminary lemmas
Lemma .(see []) Letη,ηbe two arbitrary complex numbers,and f(z)be a
meromor-phic function of finite orderσ.Letε> be given,then there exists a set E⊂(, +∞)with finite logarithmic measure such that for all r∈/E∪[, ],we have
exp–rσ–+ε≤f(z+η) f(z+η)
≤exprσ–+ε.
Remark . It follows from Lemma . that
fi(fz()z)= i
j=C
j
i(–)i–jf(z+jc)
f(z)
≤
i
j=
Cijf(z+jc) f(z)
≤
i
j=
Cijexprσ–+ε= iexprσ–+ε, i∈N+.
r∈/[, ]∪E,r→ ∞,we have
exp–rβ+ε≤f(z)≤exprβ+ε.
Lemma .(see []) Suppose that P(z) = (α+βi)zn+· · · (α,β are real numbers such
that|α|+|β| = )is a polynomial with degree n(≥),andω(z) (≡)is a meromorphic function withσ(ω) <n.Set g(z) =ω(z)eP(z),z=reiθ,δ(P,θ) =αcosnθ –βsinnθ.Then for
any givenε> ,there exists a set H⊂[, π)with linear measure zero,such that for any θ∈[, π)\(H∪H),there exists r=r(θ,ε) (> )such that for|z|=r>r,we have
(i) ifδ(P,θ) > ,then
exp( –ε)δ(P,θ)rn<greiθ<exp( +ε)δ(P,θ)rn;
(ii) ifδ(P,θ) < ,then
exp( +ε)δ(P,θ)rn<greiθ<exp( –ε)δ(P,θ)rn,
where H={θ∈[, π) :δ(P,θ) = }is a finite set.
Remark . LetP(z) =azn+· · · be a polynomial with degreenandz=reiθ, we denote δ(P,θ) =|a|cos(arga+nθ).
Lemma . (see []) Let f(z)be a nonconstant meromorphic function, ε> ,and c∈
C\{}.Ifζ=σ(f) < ,then
m r,f(z+c)
f(z)
=o T(r,f) r–ζ–ε
for all r outside of a set of finite logarithmic measure.
Lemma . Let cj,j= , . . . ,k be distinct nonzero complex constants,Aj(z),j= , , . . . ,k,
F(z)be meromorphic functions such that Ak(z)A(z)F(z)≡.If f(z)is a meromorphic
so-lution of the difference equation
Ak(z)f(z+ck) +· · ·+A(z)f(z+c) +A(z)f(z) =F(z) (.)
and satisfiesmax{σ(F),σ(Aj),j= , , . . . ,k}=ω<σ(f) =σ ( <σ≤ ∞),σ(f) < ,then we
have
λ(f) =σ(f).
Proof We use a similar proof to the one in [] here. We rewrite (.) as
f(z)=
F(z) Ak(z)
f(z+ck)
f(z) +· · ·+A(z)
f(z+c)
f(z) +A(z)
. (.)
By Lemma ., there exists a setE⊂(, +∞) of finite logarithmic measure such that for allzsatisfying|z|=r∈/E, we have
m r,f(z+cj)
f(z)
hold. Then, for sufficiently largern, we have It follows from (.) that for sufficiently largern,
T(rn,f)≤N rn,
3 Proofs of Theorems 1.1 and 1.2
Forj∈I, there exists sufficiently smallε (> ) such that for allθ∈(θ–ε,θ+ε)⊂ (, π), we have
cos(argain+nθ) >max j∈I
cos(argajn+nθ),cos(argbjn+nθ),
.
Since|ain| ≥ |ajn| ≥ |bjn|> ,j=i, we see that
δ(Pi,θ) >max j∈I
δ(Pj,θ),δ(Qj,θ),
.
Forj∈I, there exists sufficiently smallε(> ) such that for allθ ∈(θ–ε,θ+ε)⊂ (, π), we have
cos(argain+nθ) =cos(argbjn+nθ) >max j∈I
cos(argajn+nθ),
.
Since|ain|>|bjn|,j=i, and|ain| ≥ |ajn|, we see that
δ(Pi,θ) >max j∈I
δ(Pj,θ),δ(Qj,θ)
> .
Forj∈I, there exists sufficiently smallε(> ) such that for allθ ∈(θ–ε,θ+ε)⊂ (, π), we have
cos(argain+nθ) =cos(argajn+nθ) >max j∈I
cos(argbjn+nθ),
.
Since|ain|>|ajn| ≥ |bjn|,j=i, we see that
δ(Pi,θ) >max j∈I
δ(Pj,θ),δ(Qj,θ)
> .
Forj∈I, there exists sufficiently smallε(> ) such that for allθ∈(θ–ε,θ+ε)⊂ (, π), we have
cos(argain+nθ) =cos(argbjn+nθ) =cos(argajn+nθ) > .
Since|ain|>|ajn| ≥ |bjn|, we see that
δ(Pi,θ) >max j∈I
δ(Pj,θ),δ(Qj,θ)
> .
Setε=min{ε,ε,ε,ε}, then for anyθ∈(θ–ε,θ+ε)⊂(, π), we have
δ(Pi,θ) =δ>max
j∈I
δ(Pj,θ),δ(Qj,θ),
=δ. (.)
If|bin|<|ain|andargbin=argain, then for allθ∈(θ–ε,θ+ε)⊂(, π), we have
Equation (.) gives
Then from (.), for sufficiently larger, we have
Sinceσ(ϕ) <n+ ,ϕ(z) (≡) does not solve (.), that is,
Ak(z)ϕ(z+ck) +· · ·+A(z)ϕ(z+c) +A(z)ϕ(z)≡
and
σAk(z)ϕ(z+ck) +· · ·+A(z)ϕ(z+c) +A(z)ϕ(z)
≤maxσ(Aj),j= , , . . . ,k,σ(ϕ)
<n+ ≤σ(f).
Therefore, by Lemma ., we haveλ(g) =σ(g),i.e.,λ(f–ϕ) =σ(f).
The proof of Theorem . is completed.
Proof of Theorem. Equation (.) gives
–A(z) =Ak(z)
kf(z)
f(z) +· · ·+A(z) f(z)
f(z) . (.)
By Remark ., we have
fj(fz()z)≤Oexprσ–+ε, j= , . . . ,k. (.)
By combining (.), (.), (.) with (.), for allz=reiθ, whereargz=θ∈(θ
–ε,θ+ ε)\(H∪H),|z|=r∈/[, ]∪E∪Eand for sufficiently larger, we have
exp( –ε)δrn
– exp( +ε)δrn
exprω+ε
≤Oexprσ–+εexp( +ε)δrn
exprω+ε.
By using a similar method to the one in the proof of Theorem ., we haveσ(f) =σ≥n+ . Further, setg(z) =f(z) –ϕ(z), theng(z) solves the equation
Ak(z)kg(z) +· · ·+A(z)g(z) +A(z)g(z)
= –Ak(z)kϕ(z) –· · ·–A(z)ϕ(z) –A(z)ϕ(z).
Sinceσ(ϕ) <n+ ,ϕ(z) (≡) does not solve (.), that is,
Ak(z)kϕ(z) +· · ·+A(z)ϕ(z) +A(z)ϕ(z)≡
and
σAkkϕ+· · ·+Aϕ+Aϕ
≤maxσ(Aj),j= , , . . . ,k,σ(ϕ)
<n+ ≤σ(f).
Therefore, by Lemma . and Remark ., we haveλ(g) =σ(g),i.e.,λ(f–ϕ) =σ(f).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors drafted the manuscript, and they read and approved the final manuscript.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11301233, 11171119), the Natural Science Foundation of Jiangxi Province in China (20151BAB201004), and the Youth Science Foundation of Education Bureau of Jiangxi Province in China (GJJ14271).
Received: 2 February 2015 Accepted: 24 April 2015 References
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