R E S E A R C H
Open Access
A discrete Hilbert-type inequality in the
whole plane
Dongmei Xin
1*, Bicheng Yang
1and Qiang Chen
2*Correspondence:
1Department of Mathematics,
Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China
Full list of author information is available at the end of the article
Abstract
By the use of weight coefficients and technique of real analysis, a discrete Hilbert-type inequality in the whole plane with multi-parameters and a best possible constant factor is given. The equivalent forms, the operator expressions, and a few particular inequalities are considered.
MSC: 26D15; 47A05
Keywords: Hilbert-type inequality; weight coefficient; equivalent form; operator expression
1 Introduction
Suppose that p> , p + q = , am,bn≥, a={am}m∞= ∈lp, b={bn}∞n= ∈ lq, ap =
(∞m=apm)
p> ,b
q> . We have the following well-known Hardy-Hilbert inequality:
∞
n= ∞
m= ambn m+n<
π
sin(πp)apbq, () where the constant factor sin(ππ/p) is the best possible (cf.[]). Also we have the following Hilbert-type inequality:
∞
n= ∞
m=
ambn
max{m,n}<pqapbq, ()
with the best possible constant factorpq(cf.[]). Inequalities () and () are important in analysis and its applications (cf.[–]).
In , Yang gave an extension of () as follows (cf.[]): If <λ,λ≤,λ+λ=λ, am,bn≥, <ap,ϕ={
∞
m=mp(–λ)–a p m}
p<∞, <b
q,ψ={
∞
n=nq(–λ)–b q n}
q <∞,
then
∞
n= ∞
m=
ambn
(max{m,n})λ< λ λλ
ap,ϕbq,ψ, ()
where the constant factor λ
λλ is the best possible.
Forλ= ,λ=q,λ=p, inequality () reduces to (). Some other results relate to ()-()
are provided by [–].
In this paper, by the use of weight coefficients and the technique of real analysis, an extension of () in the whole plane is given as follows: For <λ,λ ≤,λ+λ=λ, am,bn≥, <
∞
|m|=|m|p(–λ)–a p
m<∞, <∞|n|=|n|q(–λ)–b q
n<∞, we have
∞
|n|= ∞
|m|=
(max{|m|,|n|})λambn
< λ λλ
∞
|m|=
|m|p(–λ)–ap
m
p∞
|n|=
|n|q(–λ)–bq
n
q
. ()
Moreover, a generation of () with multi-parameters and a best possible constant factor is proved. The equivalent forms, the operator expressions and a few particular inequalities are also considered.
2 Some lemmas
In the following, we agree that N ={, , . . .}, p> , p +q = ,α,β ∈(,π),λ,λ> –η,
λ+λ=λ, and for|x|,|y|> ,
k(x,y) := (min{|x|+xcosα,|y|+ycosβ}) η
(max{|x|+xcosα,|y|+ycosβ})λ+η. ()
Lemma (cf.[]) Suppose that g(t) (> )is decreasing inR+and strictly decreasing in
[n,∞) (n∈N),satisfying
∞
g(t)dt∈R+.We have
∞
g(t)dt<
∞
n= g(n) <
∞
g(t)dt. ()
Definition Define the following weight coefficients:
ω(λ,m) := ∞
|n|=
k(m,n)(|m|+mcosα) λ
(|n|+ncosβ)–λ, |m| ∈N, ()
(λ,n) := ∞
|m|=
k(m,n) (|n|+ncosβ) λ
(|m|+mcosα)–λ, |n| ∈N, ()
where∞|j|=· · ·=–j=–∞ · · ·+∞j=· · · (j=m,n).
Lemma Ifλ≤ –η,then for kβ(λ) :=(λ+η)csc β
(λ+η)(λ+η),we have
kβ(λ)
–θ(λ,m) <ω(λ,m) <kβ(λ), |m| ∈N, ()
where
θ(λ,m) :=
(λ+η)(λ+η)
λ+ η
+cosβ |m|+mcosα
(min{,u})ηuλ–
(max{,u})λ+η du =O
(|m|+mcosα)η+λ
Proof For|x|> , we set
k()(x,y) := [min{|x|+xcosα,y(cosβ– )}] η
[max{|x|+xcosα,y(cosβ– )}]λ+η, y< ,
k()(x,y) := [min{|x|+xcosα,y( +cosβ)}] η
[max{|x|+xcosα,y( +cosβ)}]λ+η, y> , from which we have
k()(x, –y) = [min{|x|+xcosα,y( –cosβ)}] η
[max{|x|+xcosα,y( –cosβ)}]λ+η, y> . We obtain
ω(λ,m) = –∞
n=–
k()(m,n)(|m|+mcosα) λ
[n(cosβ– )]–λ
+
∞
n=
k()(m,n)(|m|+mcosα) λ
[n( +cosβ)]–λ
=(|m|+mcosα) λ
( –cosβ)–λ
∞
n=
k()(m, –n) n–λ
+(|m|+mcosα) λ
( +cosβ)–λ
∞
n=
k()(m,n)
n–λ . ()
For fixed|m| ∈N,λ≤ –η, we find that
k()(m, –y) y–λ =
[min{|m|+mcosα,y( –cosβ)}]η
y–λ[max{|m|+mcosα,y( –cosβ)}]λ+η
=
⎧ ⎨ ⎩
(–cosβ)η
(|m|+mcosα)λ+ηy–(λ+η), <y<
|m|+mcosα
–cosβ ,
(|m|+mcosα)η
(–cosβ)λ+η y+(λ+η), y≥
|m|+mcosα
–cosβ
is decreasing fory> and strictly decreasing fory≥|m|–+mcoscosβα. Under the same assump-tions, it is evident that
k()(m,y)
y–λ =
[min{|m|+mcosα,y( +cosβ)}]η
y–λ[max{|m|+mcosα,y( +cosβ)}]λ+η
=
⎧ ⎨ ⎩
(+cosβ)η
(|m|+mcosα)λ+ηy–(λ+η), <y<
|m|+mcosα
+cosβ ,
(|m|+mcosα)η
(+cosβ)λ+η y+(λ+η), y≥|m| +mcosα
+cosβ
is decreasing fory> and strictly decreasing fory≥|m|++mcoscosβα. By () and (), we have
ω(λ,m) <
(|m|+mcosα)λ
( –cosβ)–λ
∞
k()(m, –y) y–λ dy
+(|m|+mcosα) λ
( +cosβ)–λ
∞
Settingu=|m|y(–+mcoscosβ)α(|m|y(++mcoscosβ)α) in the above first (second) integral, by simplifications, we find
ω(λ,m) <
–cosβ +
+cosβ
∞
(min{,u})ηuλ–
(max{,u})λ+η du = cscβ
uη+λ–du+
∞
uλ–
uλ+η du
=(λ+ η)csc
β
(λ+η)(λ+η)
=kβ(λ).
Still by () and (), we have
ω(λ,m) >
(|m|+mcosα)λ
( –cosβ)–λ
∞
k()(m, –y) y–λ dy
+(|m|+mcosα) λ
( +cosβ)–λ
∞
k()(m,y)
y–λ dy
≥
–cosβ
∞
+cosβ |m|+mcosα
(min{,u})ηuλ–
(max{,u})λ+η du
+
+cosβ
∞
+cosβ |m|+mcosα
(min{,u})ηuλ–
(max{,u})λ+η du =kβ(λ)
–θ(λ,m) > .
We obtain for|m|+mcosα≥ +cosβ
<θ(λ,m) =
(λ+η)(λ+η)
λ+ η
+cosβ |m|+mcosα
(min{,u})ηuλ–
(max{,u})λ+η du =(λ+η)(λ+η)
λ+ η
+cosβ |m|+mcosα
uη+λ–du
= λ+η λ+ η
+cosβ |m|+mcosα
η+λ
.
Then we have () and ().
In the same way, we have the following.
Lemma Ifλ≤ –η,then for kα(λ) =(λ+η)csc α
(λ+η)(λ+η),we have
kα(λ)
–ϑ(λ,n) <(λ,n) <kα(λ), |n| ∈N, ()
where
ϑ(λ,n) :=
(λ+η)(λ+η)
λ+ η
+cosα |n|+ncosβ
(min{,u})ηuλ–
(max{,u})λ+η du =O
(|n|+ncosβ)η+λ
Lemma Ifθ∈(,π),then forρ> ,Hρ(θ) :=
∞
|n|=(|n|+ncos θ)+ρ,we have
Hρ(θ) =
( +cosθ)+ρ +
( –cosθ)+ρ
+ρO() ρ
ρ→+ . ()
Proof We have
Hρ(θ) =
–∞
n=–
[n(cosθ– )]+ρ +
∞
n=
[n(cosθ+ )]+ρ
=
( –cosθ)+ρ +
( +cosθ)+ρ
∞
n=
n+ρ.
By (), we find
Hρ(θ) =
( –cosθ)+ρ +
( +cosθ)+ρ
+
∞
n=
n+ρ
<
( –cosθ)+ρ+
( +cosθ)+ρ
+
∞
dy y+ρ
= ρ
( –cosθ)+ρ+
( +cosθ)+ρ
( +ρ),
Hρ(θ) >
( –cosθ)+ρ+
( +cosθ)+ρ
∞
dy y+ρ
= ρ
( –cosθ)+ρ+
( +cosθ)+ρ
.
Hence we have ().
3 Main results
Theorem Ifλ,λ≤ –η,am,bn≥ (|m|,|n| ∈N),
<
∞
|m|=
|m|+mcosα p(–λ)–apm<∞,
<
∞
|n|=
|n|+ncosβ q(–λ)–bqn<∞,
kα,β(λ) :=k
p β(λ)k
q α(λ) =
(λ+ η)cscpβcscqα (λ+η)(λ+η)
, ()
then we have the following equivalent inequalities:
I:=
∞
|n|= ∞
|m|=
k(m,n)ambn
<kα,β(λ)
∞
|m|=
|m|+mcosα p(–λ)–apm
p
×
∞
|n|=
|n|+ncosβ q(–λ)–bqn
q
J:=
∞
|n|=
|n|+ncosβ pλ–
∞
|m|=
k(m,n)am pp
<kα,β(λ)
∞
|m|=
|m|+mcosα p(–λ)–apm
p
. ()
In particular,forα=β=π,we have the following equivalent inequalities:
∞ |n|= ∞ |m|=
(min{|m|,|n|})η (max{|m|,|n|})λ+ηambn
< (λ+ η) (λ+η)(λ+η)
∞
|m|=
|m|p(–λ)–ap
m
p∞
|n|=
|n|q(–λ)–bq
n q , () ∞ |n|=
|n|pλ–
∞
|m|=
(min{|m|,|n|})η (max{|m|,|n|})λ+ηam
pp
< (λ+ η) (λ+η)(λ+η)
∞
|m|=
|m|p(–λ)–ap
m
p
. ()
Proof By Hölder’s inequality (cf.[]) and (), we have
∞
|m|=
k(m,n)am p
=
∞
|m|=
k(m,n)(|m|+mcosα)
(–λ)/qa
m
(|n|+ncosβ)(–λ)/p
(|n|+ncosβ)(–λ)/p
(|m|+mcosα)(–λ)/q p
≤
∞
|m|=
k(m,n)(|m|+mcosα)
(–λ)p/q
(|n|+ncosβ)–λ a
p m × ∞ |m|=
k(m,n)(|n|+ncosβ)
(–λ)q/p
(|m|+mcosα)–λ p–
= ((λ,n))
p–
(|n|+ncosβ)pλ–
∞
|m|=
k(m,n)(|m|+mcosα)
(–λ)p/q
(|n|+ncosβ)–λ a
p m.
By (), we have
J<k
q α(λ)
∞ |n|= ∞ |m|=
k(m,n)(|m|+mcosα)
(–λ)(p–)
(|n|+ncosβ)–λ a
p m p =k q α(λ)
∞ |m|= ∞ |n|=
k(m,n)(|m|+mcosα)
(–λ)(p–)
(|n|+ncosβ)–λ a
p m p =k q α(λ)
∞
|m|=
ω(λ,m)
|m|+mcosα p(–λ)–apm
p
. ()
By Hölder’s inequality (cf.[]), we have
I =
∞
|n|=
|n|+ncosβ λ–
p
∞
|m|=
k(m,n)am
|n|+ncosβ
p–λ bn
≤J
∞
|n|=
|n|+ncosβ q(–λ)–bqn
q
. ()
Then by (), we have ().
On the other hand, assuming that () is valid, we set
bn:=
|n|+ncosβ pλ–
∞
|m|=
k(m,n)am p–
, |n| ∈N.
Then it follows that
J=
∞
|n|=
|n|+ncosβ q(–λ)–bqn
p .
By (), we findJ<∞. IfJ= , then () is evidently valid; ifJ> , then by (), we have
<
∞
|n|=
|n|+ncosβ q(–λ)–
bqn=Jp=I
<kα,β(λ)
∞
|m|=
|m|+mcosα p(–λ)– apm
p
×
∞
|n|=
|n|+ncosβ q(–λ)– bqn
q ,
J=
∞
|n|=
|n|+ncosβ q(–λ)–bqn
p
<kα,β(λ)
∞
|m|=
|m|+mcosα p(–λ)–apm
p ,
namely, () follows, which is equivalent to ().
Theorem As regards the assumptions of Theorem,the constant factor kα,β(λ)in() and()is the best possible.
Proof For anyε∈(,q(λ+η)), we setλ=λ+εq (> –η),λ=λ–εq (∈(–η, –η)), and
am:=
|m|+mcosα (λ–
ε
p)–=|m|+mcosα λ–ε– |
m| ∈N ,
bn:=
|n|+ncosβ (λ–
ε
Then by () and (), we find
I:=
∞
|m|=
|m|+mcosα p(–λ)– apm
p
×
∞
|n|=
|n|+ncosβ q(–λ)– bqn
q
=
∞
|m|=
(|m|+mcosα)+ε
p∞
|n|=
(|n|+ncosβ)+ε
q
= ε
( +cosα)+ε+
( –cosα)+ε
p
+εO()
p
×
( +cosβ)+ε+
( –cosβ)+ε
q
+εO()
q,
I:=
∞
|n|= ∞
|m|=
k(m,n)ambn
=
∞
|m|= ∞
|m|=
k(m,n)(|m|+mcosα) λ–ε–
(|n|+ncosβ)–λ
=
∞
|m|=
ω(λ,m)
(|m|+mcosα)ε+ ≥kβ(λ) ∞
|m|=
–θ(λ,m)
(|m|+mcosα)ε+
=kβ(λ)
∞
|m|=
(|m|+mcosα)ε+ – ∞
|m|=
O((|m|+mcosα)(εp+λ+η)+)
=kβ(λ) ε
( +cosα)+ε +
( –cosα)+ε
+εO() –εO()
.
If there exists a constantk≤kα,β(λ), such that () is valid when replacingkα,β(λ) by
k, then in particular, we haveεI<εkI, namely,
kβ(λ)
( +cosα)+ε+
( –cosα)+ε
+εO() –εO()
<k
( +cosα)+ε +
( –cosα)+ε
p
+εO()
p
×
( +cosβ)+ε+
( –cosβ)+ε
q
+εO()
q.
It follows that
(λ+ η) (λ+η)(λ+η)
namely,
kα,β(λ) =
(λ+ η)cscpβcscqα (λ+η)(λ+η) ≤
k.
Hence,k=kα,β(λ) is the best possible constant factor of ().
The constant factorkα,β(λ) in () is still the best possible. Otherwise, we would reach
a contradiction by () that the constant factor in () is not the best possible.
4 Operator expressions
We set functions(m) and(n) as follows:
(m) :=|m|+mcosα p(–λ)– |m| ∈N , (n) :=|n|+ncosβ q(–λ)– |n| ∈N ,
from which we have
–p(n) =|n|+ncosβ pλ– | n| ∈N .
We also set the following weight normed spaces:
lp,:=
a={am}∞|m|=;ap,=
∞
|m|=
(m)|am|p
p <∞
,
lq,:=
b={bn}∞|n|=;bq,=
∞
|n|=
(n)|bn|q
q <∞
,
lp,–p:=
c={cn}∞|n|=;cp,–p=
∞
|n|=
–p(n)|cn|p
p <∞
.
Then fora={am}∞|m|=∈lp,,c={cn}∞|n|=,cn=
∞
|m|=k(m,n)am, in view of (), we have
cp,–p<kα,β(λ)ap,<∞, namely,c∈lp,–p.
Definition Define a Hilbert-type operatorT :lp, →lp,–p as follows: For any a= {am}∞|m|=∈lp,, there exists a unique representationc=Ta∈lp,–p. We also define the formal inner product ofTaandb={bn}∞|n|=∈lq,(bn≥) as follows:
(Ta,b) :=
∞
|n|= ∞
|m|=
k(m,n)ambn. ()
Then foram≥ (|m| ∈N), we may rewrite () and () as follows:
(Ta,b) <kα,β(λ)ap,bq,, ()
We define the norm of operatorTas follows:
T:= sup
a(=θ)∈lp,
Tap,–p ap,
. ()
ThenTap,–p≤ T · ap,. Since by Theorem , the constant factorkα,β(λ) in ()
is the best possible, we have
T=kα,β(λ) =
(λ+ η)cscpβcscqα (λ+η)(λ+η)
. ()
Remark (i) Forη= , () reduces to the following inequality:
∞
|n|= ∞
|m|=
(max{|m|+mcosα,|n|+ncosβ})λambn
< λ λλ
cscpβcscqα
∞
|m|=
|m|+mcosα p(–λ)– apm
p
×
∞
|n|=
|n|+ncosβ q(–λ)– bqn
q
. ()
In particular, forα=β=π
, () reduces to (). Ifa–m=am,b–n=bn(m,n∈N), then ()
reduces to (). Hence, () is an extension of () with multi-parameters. (ii) Forη= –λ, –≤λ,λ< in (), we have
∞
|n|= ∞
|m|=
(min{|m|+mcosα,|n|+ncosβ})λambn
<(–λ) λλ
cscpβcscqα
∞
|m|=
|m|+mcosα p(–λ)–apm
p
×
∞
|n|=
|n|+ncosβ q(–λ)–bqn
q
. ()
In particular, forα=β=π, we have
∞
|n|= ∞
|m|=
(min{|m|,|n|})λambn
<(–λ) λλ
∞
|m|=
|m|p(–λ)–ap
m
p∞
|n|=
|n|q(–λ)–bq
n
q
(iii) Forλ= in (), we haveλ= –λ,|λ|<η(η> ) and
∞
|n|= ∞
|m|=
min{|m|+mcosα,|n|+ncosβ}
max{|m|+mcosα,|n|+ncosβ}
η
ambn
< η η–λ
cscpβcscqα
∞
|m|=
|m|+mcosα p(–λ)–apm
p
×
∞
|n|=
|n|+ncosβ q(+λ)–bqn
q
. ()
In particular, forα=β=π, we have
∞
|n|= ∞
|m|=
min{|m|,|n|}
max{|m|,|n|} η
ambn
< η η–λ
∞
|m|=
|m|p(–λ)–ap
m
p∞
|n|=
|n|q(–λ)–bq
n
q
. ()
The above particular inequalities are all with the best possible constant factors.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BY carried out the mathematical studies, participated in the sequence alignment and drafted the manuscript. DX and QC participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China. 2Department of Computer Science, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China.
Acknowledgements
This work is supported by the Science and Technology Planning Project of Guangdong Province (No. 2013A011403002), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25).
Received: 6 March 2016 Accepted: 19 April 2016
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