R E S E A R C H
Open Access
Some results of certain types of difference
and differential equations
Yong Liu
2, Yin Hong Cao
1*and Hong Xun Yi
2*Correspondence: [email protected] 1School of Mathematics and Information Sciences, Henan Polytechnic University, Jiaozuo, Henan 454000, P.R. China Full list of author information is available at the end of the article
Abstract
In this article, we shall utilize the value distribution theory and complex oscillation theory to investigate certain types of difference and differential equations. The results we obtain generalize some previous results of Gundersen and Yang.
MSC: 30D35; 34M10
Keywords: entire functions; uniqueness; complex difference; share value
1 Introduction and main results
In this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna theory (e.g., see [–]). In addition, we will use the notationσ(f) to denote the order of the meromorphic functionf(z). We recall the defini-tion of hyper-order (see []),σ(f) off(z) is defined by
σ(f) =lim sup
r→∞
log logT(r,f)
logr .
Letfandgbe two non-constant meromorphic functions in the complex plane. ByS(r,f), we denote any quantity satisfyingS(r,f) =o(T(r,f)) asr→ ∞, possibly outside a set ofr with finite linear measure. Then the meromorphic functionβis called a small function off, ifT(r,β) =S(r,f). Iff –βandg–βhave the same zeros, counting multiplicity (ig-noring multiplicity), then we sayf andgshare the small functionβCM (IM).
Letzbe a zero off –βwith multiplicitypand a zero ofg–βwith multiplicityq. We
denote byNL(r,f–β ) the counting function of the zeros off–βwherep>q≥, each point
countedp–qtimes. In the same way, we also defineNL(r,g–β ).
Letf(z) be transcendental meromorphic function in the plane,c∈C\ {}be a constant such thatf(z)≡f(z+c). The forward differencesnf(z) are defined in the standard way [] by
f(z) =f(z+c) –f(z), n+f(z) =nf(z+c) –nf(z), n= , , . . . .
In ,Brückraised the following conjecture:
Conjecture Letf be a non-constant entire function such that the hyper orderσ(f) <∞
andσ(f) is not a positive integer. Iff andfshare the finite valueaCM, then
f–a f–a =c,
wherecis a nonzero constant.
The case thata= had been proved byBrückhimself in []. From differential equations
f– f– =e
zn , f
–
f– =e
ez ,
we see that when the hyper orderσ(f) off is a positive integer or infinite, the conjecture
ofBrückdoes not hold.
Gundersen and Yang proved the conjecture holds for entire functions of finite order, see [], and Yang generalized this finite order forf(k)(k≥), instead off, see []. Chen
and Shon proved the conjecture holds if σ(f) < , see []. In terms of sharing a small
functionαIM, recently Wang has generalized Gundersen and Yang’s results, see []. In this paper, we consider the uniqueness of entire functions sharing a small function with their linear difference and differential polynomial. Now we present the main theo-rems.
Theorem . Let f(z)be a non-constant entire function,σ(f)(<∞) is not a positive
in-teger. Set L(f) =ak(z)f(k)(z) +ak–(z)f(k–)(z) +· · ·+a(z)f
(z) +f(z)(k≥), where aj(z)
(≤j≤k) are entire functions of order less than and ak(z)≡. If f(z)and L(f)share z
IM, and
s:=max
lim sup
r→∞
log(NL(r,f(z)–z))
logr ,lim supr→∞
log(NL(r,L(f)–z))
logr
< , (.)
then L(f) –z=h(z)(f(z) –z), where h(z)is a meromorphic function of order no greater
than s.
Remark Note that the termf(z) cannot be contained inL(f), otherwise Theorem .
does not hold. For example: Setf(z) = e–z+zandL
(f) =f(z) + f(z) +f(z). Thenf(z)
andL(f) sharezIM, but
L(f) –z
f(z) –z =e
z.
Theorem . Let f(z)be a non-constant entire function,σ(f)(<∞) is not a positive
inte-ger. Set L(f) =ak(z)f(k)(z) +ak–(z)f(k–)(z) +· · ·+a(z)f
(z) +a(z)f(z)(k≥), where aj(z)
(≤j≤k) are entire functions andmax{σ(aj)|j= , , . . . ,k}<σ(a)∈N. If f(z)and L(f)
share a IM (a is a constant), then
L(f) –a=h(z)
f(z) –a,
Theorem . Let f(z)be a non-constant entire function of order less than and a(z)be a non-zero small function of f(z). Set A(f) =ak(z)kf(z) +· · ·+a(z)f(z) +a(z)f(z), where
aj(z)(j= , , . . . ,k) are polynomial and ak(z)≡. If f(z) –a(z) = →A(f) –a(z) = , then
A(f) –a(z) f(z) –a(z) =B(z),
where B(z)is a non-zero polynomial.
2 Some lemmas
In order to prove our theorems, we need the following lemmas and notions.
Following Hayman [, pp.-], we define anε-set to be a countable union of open discs not containing the origin and subtending angles at the origin whose sum is finite. IfEis anε-set then the set ofr≥ for which the circleS(,r) ={z∈C:|z|=r}meetsE has finite logarithmic measure, and for almost all realθ the intersection ofEwith the ray
argz=θis bounded.
Lemma .([]) Let n∈N. Let f(z)be transcendental and meromorphic of order less than in the plane. Then there exists anε-set Ensuch that
nf(z)→f(n)(z) as z→ ∞inC\En.
Lemma . ([]) Let w(z) be an entire function of order ρ(w) = β <
, A(r) = inf|z|=rlog|w(z)|and B(r) =sup|z|=rlog|w(z)|. Ifβ<α< , then
log densA(r) >cos(π α)B(r)(E)≥ –β α,
where the lower logarithmic densitylog densH of subset H⊂(,∞)is defined by
log densH=lim inf
r→∞
r
χH(t)/tdtlogr,
and the upper logarithmic densitylog densH of subset H⊂(,∞)is defined by
log densH=lim sup
r→∞ r
χH(t)/tdtlogr,
whereχH(t)is the characteristic function of the set H.
Lemma .([]) Let f(z)be an entire function of finite order. Suppose thatαis a non-zero
small function of f(z). Then there exists a set E⊂(,∞)satisfyinglog dens(E) = , such that
log+M(r,α) log+M(r,f) →,
M(r,α) M(r,f)→,
Lemma .([]) Let f(z)be a meromorphic function withρ(f) =η<∞. Then for any givenε> , there is a set E⊂(, +∞)that has finite logarithmic measure such that
f(z)≤exprη+ε,
holds for|z|=r∈/[, ]∪E, r→ ∞.
Applying Lemma . tof, it is easy to see that for any givenε> , there is a setE⊂(,∞)
of finite logarithmic measure such that
exp–rη+ε≤f(z)≤exprη+ε
holds for|z|=r∈/[, ]∪E,r→ ∞.
Lemma .([]) Let f(z)be a transcendental meromorphic function, andα> be a given constant. Then
(i) there exists a setE⊂(,∞)with finite linear measure zero and a constantB> that depends only onαandj= , . . . ,k, such that ifϕ∈[, π)\E, then there is a constantR=R(ϕ) > so that for allzsatisfyingargz=ϕand|z|=r≥R, we have
ff(j()(zz))≤B
T(αr,f) r
logαrlogT(αr,f) j
, (.)
for allj= , . . . ,k;
(ii) there exists a setE⊂(,∞)with finite logarithmic measure and a constantB>
that depends only onαandj= , . . . ,k, such that for allzsatisfying|z|=r∈/[, ]∪E, we have (.) holds.
Lemma .([]) Let f(z)be an entire function of infinite order withσ(f) =σ, andμ(r)
be the central index of f(z). Then
lim sup
r→∞
log logμ(r)
logr =σ(f) =σ.
3 Proof of Theorem 1.1
Under the hypothesis of Theorem ., see [], it is easy to get that
L(f) –z
f(z) –z = h(z)
h(z)
eθ(z), (.)
whereθ(z) is an entire function, entire functionsh(z) andh(z) satisfy
σ(h) =lim sup
r→∞
log(NL(r,L(f)–z))
logr , σ(h) =lim supr→∞
log(NL(r,f(z)–z))
logr .
Therefore, by (.), we see thath(z) =h(z)
h(z) is a meromorphic function of order no greater
thans(< ). Iff(z) is a polynomial, thenσ(L(f)–z
f(z)–z) = . Theorem . holds under this
Substitutingf(z) =F(z) +zinto (.), we get scendental entire function, rewritten (.), we have
ak(z)
By Wiman-Valiron theory, there exists a setE⊂(,∞) having finite logarithmic
whereυ(r) is the central index ofF(z). By (.), (.), (.), we obtain
By combining (.) and (.), we obtain
σ(f) =σ(F) =σ
this contradicts the hypothesis of Theorem .. Theorem . is thus proved.
4 Proof of Theorem 1.2
Under the hypothesis of Theorem ., it is easy to get that
L(f) –a
Proof: By the Hadamard factorization theorem, we have
A(f) –a(z)
whereB(z) is an entire function of order less than . Iff(z) is polynomial, Theorem . holds. Next, we consider thatf(z) is transcendental. SetF(z) =f(z) –a(z). Then F(z) is transcendental, andσ(F) <. Substitutingf(z) =F(z) +a(z) into (.), we have
ak(z)kF(z) +ak–(z)k–F(z) +· · ·+a(z)F(z) +b(z)
F(z) =B(z), (.)
whereb(z) =ak(z)ka(z) +· · ·+a(z)a(z) +a(z)(a(z) – ). By Lemma ., there exists an
ε-setEnsuch that
jF(z)∼F(j)(z) (j= , , . . . ,n), (.)
asz→ ∞inC\En. By Wiman-Valiron theory, there is a subsetE⊂(,∞) with finite
logarithmic measure. We choosezsatisfying|z|=r∈/Eand|F(z)|=|M(r,f(z))|, then we
have
F(j)(z) F(z) =
υ(r)
r j
+o() (j= , , . . . ,k), (.)
whereυ(r) is the central index ofF(z). By (.)-(.), we have
ak(z)
υ(r)
r n
+o()+· · ·+a(z)
υ(r)
r
+o()+b(z)
F(z)=B(z) –a(z). (.)
By Lemma ., there exists a setE⊂(,∞) satisfyinglog dens(E) = such that
M(r,b(z))
M(r,F(z))→. (.)
Together withσ(F) < , we get
υ(r)
r n
≤
υ(r) r
n–
≤ · · · ≤
υ(r) r
. (.)
By (.)-(.), we have
B(z) –a(z)≤C
υ(r)
r
rC, (.)
whereCis a constant. By Lemma ., for anyαsatisfyingμ<α<, there exists a setE
withlog dens(E)≥ –μα, such that
B
reiθ–a
reiθ≥Mr, (B–a)λ
, (.)
for|z|=r∈E, whereλ=cosπ α.
Note the characteristic function ofEandEsuch that the relation
Obviously,log dens(E∪E)≤. Hence we obtain
–μ
α ≤log dens(E) +log dens(E) –log dens(E∪E)≤log dens(E∩E).
Thus, the upper logarithmic density of (E∩E)\(En∪E∪E∪[, ]) is also more than
–μα. By (.) and (.), forz∈(E∩E)\(En∪E∪E∪[, ]), we have
Mr, (B–a)λ
≤C
υ(r)
r
rC.
IfB(z) is transcendental, we getσ(f) =∞, which contradicts our assumption thatσ(f) <. SoB(z) is polynomial. This proves Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YL and YH completed the main parts of the paper. YL, YH and HX corrected the main theorems. All authors read and approved the final manuscript.
Author details
1School of Mathematics and Information Sciences, Henan Polytechnic University, Jiaozuo, Henan 454000, P.R. China. 2School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China.
Acknowledgements
The work was supported by the NNSF of China (No. 10771121), the NSF of Shangdong Province, China (No. Z2008A01) and Shandong university graduate student independent innovation fund (yzc11024).
Received: 14 May 2012 Accepted: 10 July 2012 Published: 25 July 2012 References
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doi:10.1186/1687-1847-2012-127
