R E S E A R C H
Open Access
A multidimensional Hilbert-type integral
inequality
Zhenxiao Huang
1and Bicheng Yang
2**Correspondence:
2Department 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 applying the method of weight functions and the technique of real analysis, a multidimensional Hilbert-type integral inequality with multi-parameters and the best possible constant factor related to the gamma function is given. The equivalent forms and the reverses are obtained. We also consider the operator expressions and a few particular results related to the kernels of non-homogeneous and homogeneous.
MSC: 26D15; 47A07; 37A10
Keywords: Hilbert-type integral inequality; weight function; equivalent form; Hilbert-type integral operator; gamma function
1 Introduction
Suppose thatp> ,p+q= ,f(x),g(y)≥,f ∈Lp(R
+),g∈Lq(R+),fp= (
∞
fp(x)dx)
p>
,gq> . We have the following well-known Hardy-Hilbert integral inequality (cf.[]):
∞
∞
f(x)g(y)
x+y dx dy<
π
sin(π/p)fpgq, ()
where the constant factor π
sin(π/p) is best possible. If am,bn≥,a={am}∞m=∈lp, b= {bn}∞n=∈lq,ap = (∞m=a
p
m)p > ,b
q> , then we still have the discrete variant of
the above inequality with the same best constantsin(ππ/p) as follows:
∞
m= ∞
n= ambn
m+n<
π
sin(π/p)apbq. ()
Inequalities () and () are important in the analysis and its applications (cf.[–]). In , by introducing an independent parameterλ∈(, ], Yang [] gave an extension of () atp=q= with the kernel (x+y)λ. In and , Yang [, ] gave some best
extensions of () and () as follows.
Ifλ,λ,λ∈R,λ+λ=λ,kλ(x,y) is a non-negative homogeneous function of degree
–λ, with
k(λ) =
∞
kλ(t, )tλ–dt∈R+,
φ(x) =xp(–λ)–,ψ(y) =yq(–λ)–,f(x),g(y)≥,
f ∈Lp,φ(R+) =
f;fp,φ:=
∞
φ(x)f(x)pdx
p
<∞
,
g∈Lq,ψ(R+),fp,φ,gq,ψ> , then we have the following inequality:
∞
∞
kλ(x,y)f(x)g(y)dx dy<k(λ)fp,φgq,ψ, ()
where the constant factor k(λ) is best possible. Moreover, if kλ(x,y) stays finite and kλ(x,y)xλ–(kλ(x,y)yλ–) is decreasing with respect tox> (y> ), then foram,bn≥,
a∈lp,φ=
a;ap,φ:=
∞
n=
φ(n)|an|p
p <∞
,
b={bn}∞n=∈lq,ψ,ap,φ,bq,ψ> , we have
∞
m= ∞
n=
kλ(m,n)ambn<k(λ)ap,φbq,ψ, ()
where the constant factork(λ) is still best possible.
Clearly, forλ= ,k(x,y) =x+y,λ=q,λ=p, () reduces to (), while () reduces to ().
In , Hong [] first published a multidimensional Hilbert integral inequality by using the transfer formula, which is an extension of (). Some other related results are given by [–], which provided some new methods to study these kinds of inequalities.
In this paper, by using the transfer formula and applying the method of weight func-tions and the technique of real analysis, we give a multidimensional Hilbert-type inte-gral inequality with multi-parameters and the best possible constant factor related to the gamma function. The equivalent forms and the reverses are obtained. Furthermore, we also consider the operator expressions and a few particular results related to the kernels of non-homogeneous and homogeneous.
2 Some lemmas
Ifm,n∈N(N is the set of positive integers),α,β> , we set
xα:=
m
k= |xk|α
α
x= (x, . . . ,xm)∈Rm,
yβ:=
n
k= |yk|β
β
y= (y, . . . ,yn)∈Rn.
Lemma If s∈N,γ,M> , (u)is a non-negative measurable function in(, ],and
DM:=
x∈Rs+; <u= s
i=
xi
M γ
≤
then we have the following transfer formula(cf.[]):
· · ·
DM
s
i=
xi
M γ
dx· · ·dxs
=M
ss( γ)
γs(s
γ)
(u)uγs–du, ()
where(·)is the gamma function defined by
(t) :=
∞
e–vvt–dv (t> ).
In view of (), since Rs
+=limM→∞DM, we have
· · ·
Rs+
s
i=
xi
M γ
dx· · ·dxs
= lim
M→∞
Mss( γ)
γs(s
γ)
(u)uγs–du. ()
By (), (i) for
x∈Rs+;xγ ≥
= lim
M→∞
x∈Rs+;
Mγ ≤u=
s
i=
xi
M γ
≤
,
setting (u) = (u∈(,Mγ)), it follows that
· · ·
{x∈Rs+;xγ≥}
s
i=
xi
M γ
dx· · ·dxs
= lim
M→∞
Mss( γ)
γs(s
γ)
Mγ
(u)uγs–du; ()
(ii) for
x∈Rs+;xγ ≤
= lim
M→∞
x∈Rs+; <u= s
i=
xi
M γ
≤ Mγ
,
setting (u) = (u∈(Mγ,∞)), we have
· · ·
{x∈Rs+;xγ≤}
s
i=
xi
M γ
dx· · ·dxs
= lim
M→∞
Mss( γ)
γs(s
γ)
Mγ
Remark Forδ∈ {–, },s∈N,γ,M> , settingEδ:={u> ;uδ≥ Mδγ}, in view of ()
and (), it follows that
· · ·
{x∈Rs+;xδγ≥}
s
i=
xi
M γ
dx· · ·dxs
= lim
M→∞ Mss(
γ)
γs(s
γ)
Eδ
(u)uγs–du. ()
Lemma Forδ∈ {–, },s∈N,γ,ε> ,we have
{x∈Rs+;xδγ≥}
x–γs–δεdx=
s( γ)
εγs–(s
γ)
. ()
Proof By (), forδ∈ {–, }, it follows that
{x∈Rs+;xδγ≥} x–s–δε
γ dx
=
· · ·
{x∈Rs+;xδ γ≥}
M
s
i=
xi
M
γγ–s–δε
dx· · ·dxs
= lim
M→∞ Mss(
γ)
γs(s
γ)
Eδ
Mu/γ–s–δεuγs–du
= lim
M→∞
M–δεs( γ)
γs(s
γ)
Eδ
u–γδε–du
v=Mγu
= M
–δεs( γ)
γs(s
γ)
{v>;vδ≥}
M–γv
–δε γ –M–γ
dv
=
s( γ)
γs(s
γ)
{v>;vδ≥}v
–δε
γ –dv=
s( γ)
εγs–(s
γ)
.
Hence, we have ().
Definition Form,n∈N,α,β,λ,λ> ,λ+λ=λ,η> –,δ∈ {–, },x= (x, . . . ,xm)∈
Rm+,y= (y, . . . ,yn)∈R+n, we define two weight functionsω(λ,y) and(λ,x) as follows:
ω(λ,y) :=yλβ
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
xm–δλ
α
dx, ()
(λ,x) :=xδλα
Rn+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
yn–λ
β
dy. ()
By (), we find
ω(λ,y) =yλβ
Rm+
|ln( yβ
Mδ[m i=(Mxi)α]δ/α
)|η
(max{Mδ[m
i=(
xi
M)α]δ/α,yβ})λ
×
Mm–δλ[m
i=(
xi M)α]
m–δλ
α
=yλ
β lim
M→∞
Mmm( α)
αm(m
α)
×
|ln( yβ
Mδuδ/α)| η
(max{Mδuδ/α,y β})λ
umα–du
Mm–δλu m–δλ
α
=yλ
β lim
M→∞
Mδλm( α)
αm(m
α)
|ln( yβ
Mδuδ/α)|ηu δλ
α –
(max{Mδuδ/α,y β})λ
du
v=My–δu/α
=
m( α)
αm–(m
α)
∞
|lnvδ|ηvδλ–
(max{vδ, })λdv. ()
Settingt=vδin (), forδ=±, by simplification, it follows that
ω(λ,y) =
m( α)
αm–(m
α)
∞
|lnt|ηtλ–
(max{t, })λdt
=
m( α)
αm–(m
α)
(–lnt)ηtλ–dt+
∞
(lnt)ηtλ– tλ dt
=
m( α)
αm–(m
α)
(–lnt)ηtλ–+tλ–dt
u=–lnt
=
m( α)
αm–(m
α)
∞u η
e–u(λ–)+e–u(λ–)–e–udu
=
m( α)
αm–(m
α)
∞
e–λu+e–λuuηdu
=
m( α)
αm–(m
α)
λη +
λη
∞
e–vv(η+)–dv
=
m(
α)(η+ )
αm–(m
α)
λη +
λη . ()
Lemma For m,n∈N,α,β,λ,λ,λ,λ> ,λ+λ=λ+λ=λ,η> –,δ∈ {–, },we have
ω(λ,y) =Kα(λ) :=
m(α)(η+ )
αm–(m
α)
λη +
λη
y∈Rn+, ()
(λ,x) =Kβ(λ) :=
n(β)(η+ )
βn–(n
β)
λη +
λη
x∈Rm+, ()
w(λ,y) :=y λ
β
{x∈Rm+;xδα≥}
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
xm–δλ
α dx
=Kα(λ)
–θλ(y)
, ()
θλ(y) :=
λ–η+λ–η (η+ )
y–
β
|lnt|ηtλ–
(max{t, })λdt=O
y–λ
β y∈Rn+
. ()
In view of () and (), we find
w(λ,y) =y λ
β lim
M→∞
Mδλm(
α)
αm(m
α)
{u>;uδ≥ Mδα}
|ln( yβ
Mδuδ/α)|ηu δλ
α –
(max{Mδuδ/α,y β})λ
du
v=My–δuα
=
m( α)
αm–(m
α)
{v>;vδ≥y–
β}
|lnvδ|ηvδλ–
(max{vδ, })λdv
t=vδ
=
m( α)
αm–(m
α)
∞
y–β
|lnt|ηtλ–
(max{t, })λdt
=
m( α)
αm–(m
α)
∞
|lnt|ηtλ–dt
(max{t, })λ –
y–
β
|lnt|ηtλ–dt
(max{t, })λ
=Kα(λ)
–θλ(y)
.
SettingF(u) :=u|(maxlnt|η{tt,λ–})λdt(u∈(,∞)), it follows thatF(u) is continuous in (,∞).
Since
lim
u→+
uλ/
u
|lnt|ηtλ–
(max{t, })λdt
= lim
u→+
λu(λ/)–
|lnu|ηuλ–
(max{u, })λ =ulim→+
(–lnu)ηuλ/
λ
= ,
lim
u→∞
uλ/
u
|lnt|ηtλ–
(max{t, })λdt= ,
there exists a constantL> such that
<
u
|lnt|ηtλ–
(max{t, })λdt≤Lu
λ
u∈(,∞).
Then we have
<θλ(y)≤
λ–η+λ–η (η+ ) Ly
–λ
β ,
namely,θλ(y) =O(y
–λ
β ) (y∈Rn+). Hence, we have () and ().
Lemma As the assumptions of Definition,if p∈R\{, },p+q= ,f(x) =f(x, . . . ,xm)≥ ,g(y) =g(y, . . . ,yn)≥,then(i)for p> ,we have the following inequality:
J:=
Rn+
ypλ–n
β
(ω(λ,y))p–
Rm+
|ln(yβ/xδα)| ηf(x)
(max{xδ
α,yβ})λ dx
p
dy
p
≤
Rm+
(λ,x)xαp(m–δλ)–mfp(x)dx
p
; ()
Proof (i) Forp> , by Hölder’s inequality with weight (cf.[]), it follows that
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ f(x)dx
=
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
x(m–δλ)/q
α y(n–λ)/p
β
f(x)
y(n–λ)/p
β x(m–δλ)/q
α
dx
≤
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
x(m–δλ)(p–)
α yn–λ
β
fp(x)dx
p
×
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
y(n–λ)(q–)
β xm–δλ
α
dx
q
=ω(λ,y)
qy
n p–λ
β
×
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
x(m–δλ)(p–)
α yn–λ
β
fp(x)dx
p
. ()
Then by Fubini’s theorem (cf.[]), we have
J≤
Rm+
Ri+
|ln(yβ/xδα)| η
(max{xδ
α,yβ})λ
x(m–δλ)(p–)
α yn–λ
β
fp(x)dx
dy
p
=
Rm+
Rn+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
x(m–δλ)(p–)
α yn–λ
β
dy
fp(x)dx
p
=
Rm+
(λ,x)xαp(m–δλ)–mf
p(x)dx
p
. ()
Hence, () follows.
(ii) For <p< , orp< , by the reverse Hölder inequality with weight (cf.[]), we obtain the reverse of (). Then by Fubini’s theorem, we still can obtain the reverse of
().
Lemma As the assumptions of Lemma,then(i)for p> ,we have the following inequal-ity equivalent to():
I:=
Rn+
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
f(x)g(y)dx dy
≤
Rm+
(λ,x)xαp(m–δλ)–mfp(x)dx
p
×
Rn+
ω(λ,y)yβq(n–λ)–ngq(y)dy
q
; ()
Proof (i) Forp> , by Hölder’s inequality (cf.[]), it follows that
I =
Rn+
y
n q–(n–λ) β
(ω(λ,y))
q
Rm+
|ln(yβ/xδα)| η
(max{xδ
α,yβ})λ f(x)dx
×ω(λ,y)
qy(n–λ)– n q
β g(y)
dy
≤J
Rn+
ω(λ,y)yβq(n–λ)–ngq(y)dy
q
. ()
Then by () we have ().
On the other hand, assuming that () is valid, we set
g(y) := y pλ–n
β
(ω(λ,y))p–
Rm+
|ln(yβ/xδα)|ηf(x)
(max{xδ
α,yβ})λ dx
p–
, y∈Rn+.
Then it follows that
Jp=
Rn+
ω(λ,y)yβq(n–λ)–ng
q(y)dy.
IfJ= , then () is trivially valid; ifJ=∞, then by (), () keeps the form of equality (=∞). Suppose that <J<∞. By (), we have
<
Rn+
ω(λ,y)yβq(n–λ)–ngq(y)dy=J
p
=I
≤
Rm+
(λ,x)xαp(m–δλ)–mfp(x)dx
p
×
Rn+
ω(λ,y)yq(n– λ)–n
β gq(y)dy
q
<∞.
Dividing outJp–in the above inequality, it follows that
J =
Rn+
ω(λ,y)yβq(n–λ)–ngq(y)dy
p
≤
Rm+
(λ,x)xαp(m–δλ)–mfp(x)dx
p ,
and then () follows. Hence, () and () are equivalent.
(ii) For <p< , orp< , by the same way, we have the reverse of () equivalent to the
reverse of ().
3 Main results and operator expressions
Setting functions
(x) :=xp(m–δλ)–m
α , (y) :=y
q(n–λ)–n
β
x∈Rm+,y∈Rn+,
Theorem Suppose that m,n∈N,α,β,λ,λ> ,λ+λ=λ, η> –,δ∈ {–, }, p∈
R\{, }, p+q= ,f(x) =f(x, . . . ,xm)≥,g(y) =g(y, . . . ,yn)≥,
<fp,=
Rm+
xp(m–δλ)–m
α f
p(x)dx
p <∞,
<gq, =
Rn+
yq(n–λ)–n
β g
q(y)dy
q <∞.
(i)For p> ,we have the following equivalent inequalities with the best possible constant factor K(λ):
I=
Rn+
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
f(x)g(y)dx dy<K(λ)fp,gq, , ()
J:=
Rn+
ypλ–n
β
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ f(x)dx
p
dy
p
<K(λ)fp,, ()
where we define the constant factor as follows:
K(λ) :=
Kβ(λ)
pK
α(λ)
q
=
n(
β)
βn–(n
β)
p m(
α)
αm–(m
α)
q
λη +
λη (η+ ).
(ii)For <p< ,or p< ,we still have the equivalent reverses of()and()with the same best constant factor K(λ).
Proof (i) Forp> , by the conditions, we can prove that () takes the form of strict in-equality. Otherwise, if () takes the form of equality fory∈Rn+, then there exist constants
AandB, which are not all zero, satisfying
Ax
(m–δλ)(p–)
α yn–λ
β
fp(x) =By
(n–λ)(q–)
β xm–δλ
α
a.e. inx∈Rm+. ()
IfA= , thenB= , which is impossible; ifA= , then () reduces to
xp(m–δλ)–m
α fp(x) =
Byq(n–λ)
β Axm
α
a.e. inx∈Ri
+,
which contradicts the fact that <fp,<∞. In fact, by (), it follows that
Rm+ x
–m
α dx= ∞. Hence, () takes the form of strict inequality. So does (). By () and (), we have ().
In view of () (puttingω(λ,y) = ), we still have
I≤J
Rn+
yq(n–λ)–n
β gq(y)dy
q
Then by () and (), we have (). It is evident that by Lemma and the assumptions, () and () are also equivalent.
For <ε<pλ
, we setf(x),g(y) as follows:
f(x) :=
, <xδ
α< , xδ(λ–
ε
p)–m
α , xδα≥,
g(y) :=
, <yβ< ,
y(λ– ε
q)–n
β , yβ≥.
Then, forλ=λ–εp∈(λ,λ) (⊂(,λ)), by () we find
fp,gq, =
{x∈Rm+;xδα≥}
x–m–δε α dx
p
{y∈Rn+;yβ≥} y–n–ε
β dy
q
=
ε
m(
α)
αm–(m
α)
p n(
β)
βn–(n
β)
q ,
≤
{y∈Rn+;yβ≥} y–n–ε
β O
y–
λ
β
dy
≤L
{y∈Rn+;yβ≥}
y–n–(ε+
λ )
β dy=
Ln(β) (ε+λ
)βn–(
n
β)
≤ L
n( β)
(ε+λ
)βn–(
n
β)
<∞ (L> ),
and then by () and () it follows that
I:=
Rn+
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
f(x)g(y)dx dy
=
{y∈Rn+;yβ≥} y–n–ε
β w(λ,y)dy
=Kα(λ)
{y∈Rn+;yβ≥} y–n–ε
β
–Oy–
λ
β
dy
=
εKα(λ)
n(
β)
βn–(n
β)
–εOλ() .
If there exists a constantK≤K(λ), such that () is valid when replacingK(λ) byK,
then in particular we have
m( α)
αm–(m
α)
(η+ )
λη +
λη
n( β)
βn–(n
β)
–εOλ()
≤εI<εKfp,gq, =K
m(
α)
αm–(m
α)
p n(
β)
βn–(n
β)
q ,
By the equivalency, we can prove that the constant factorK(λ) in () is best possible.
Otherwise, we would reach a contradiction by () that the constant factorK(λ) in ()
is not best possible.
(ii) For <p< , orp< , by the same way, we still can obtain the equivalent reverses of
() and () with the same best constant factor.
As the assumptions of Theorem , forp> , in view ofJ<K(λ)fp,, we give the
fol-lowing definition.
Definition We define a multidimensional Hilbert-type integral operator
T: Lp,
Rm+→Lp, –p
Rn+
as follows:
Forf ∈Lp,(Rm+), there exists a unique representationTf ∈Lp, –p(Rn+), satisfying
(Tf)(y) :=
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
f(x)dx y∈Rn+. ()
Forg∈Lq, (Rn+), we define the following formal inner product ofTf andgas follows:
(Tf,g) :=
Rn+
Rm+
|ln(yβ/xδα)|η
(max{xδ
α,yβ})λ
f(x)g(y)dx dy. ()
Then by Theorem , forp> , <fp,,gq, <∞, we have the following equivalent
inequalities:
(Tf,g) <K(λ)fp,gq, , ()
Tfp, –p<K(λ)fp,. ()
It follows thatTis bounded with
T:= sup
f(=θ)∈Lp,(Rm+)
Tfp, –p
fp, ≤ K(λ).
Since the constant factorK(λ) in () is best possible, we have
T=K(λ) =
n( β)
βn–(n
β)
p m(
α)
αm–(m
α)
q
λη +
λη (η+ ). ()
4 Some corollaries
We also set functions
(x) :=xp(m–λ)–m
α , (x) :=xp(m– λ)–m
α
x∈Rm+.
Forδ= – in Theorem , settingF(x) =xλ
Corollary Suppose that m,n∈N, α,β,λ,λ > , λ +λ =λ, η> –, p∈R\{, },
p+
q= ,F(x) =F(x, . . . ,xm)≥,g(y) =g(y, . . . ,yn)≥, <Fp,,gq, <∞. (i)For p> ,we have the following equivalent inequalities with the non-homogeneous kernel and the best possible constant factor K(λ):
Rn+
Rm+
|ln(xαyβ)|η
(max{,xαyβ})λ
F(x)g(y)dx dy<K(λ)Fp,gq, , ()
Rn+
ypλ–n
β
Rm+
|ln(xαyβ)|ηF(x)
(max{,xαyβ})λ dx
p
dy
p
<K(λ)Fp,; ()
(ii)for <p< ,or p< ,we still have the equivalent reverses of()and()with the same best constant factor K(λ).
Forδ= in Theorem , we have the following.
Corollary Suppose that m,n∈N, α,β,λ,λ> , λ+λ=λ, η> –, p∈R\{, },
p+
q= ,f(x) =f(x, . . . ,xm)≥,g(y) =g(y, . . . ,yn)≥, <fp,,gq, <∞. (i)For p> ,we have the following equivalent inequalities with the homogeneous kernel of degree
–λand the best possible constant factor K(λ):
Rn+
Rm+
|ln(yβ/xα)|η
(max{xα,yβ})λ
f(x)g(y)dx dy<K(λ)fp,gq, , ()
Rn+
ypλ–n
β
Rm+
|ln(yβ/xα)|ηf(x)
(max{xα,yβ})λ dx
p
dy
p
<K(λ)fp,; ()
(ii)for <p< ,or p< ,we still have the equivalent reverses of()and()with the same best constant factor K(λ).
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. ZH participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
Author details
1Basic Education College of Zhanjiang Normal University, Zhanjiang, Guangdong 524037, P.R. China.2Department of
Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 61370186), and 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).
Received: 2 January 2015 Accepted: 22 April 2015
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