R E S E A R C H
Open Access
An extension of a multidimensional
Hilbert-type inequality
Jianhua Zhong and Bicheng Yang
**Correspondence:
[email protected] Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China
Abstract
In this paper, by the use of weight coefficients, the transfer formula and the technique of real analysis, a new multidimensional Hilbert-type inequality with multi-parameters and a best possible constant factor is given, which is an extension of some published results. Moreover, 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; norm
1 Introduction
Ifp> , p +q = ,am,bn≥,a={am}m∞=∈lp,b={bn}∞n=∈lq,ap= (
∞
m=a
p m)
p > , bq> , then we have the following Hardy-Hilbert inequality with the best possible
con-stantsin(ππ/p): ∞
m= ∞
n= ambn
m+n< π
sin(π/p)apbq, ()
and the following Hilbert-type inequality:
∞
m= ∞
n=
ambn
max{m,n}<pqapbq ()
with the best possible constant factorpq(cf. [], Theorem , Theorem ). Inequalities () and () are important in the analysis and its applications (cf. [–]).
Assuming that{μm}∞m=,{νn}∞n=are positive sequences,
Um= m
i=
μi, Vn= n
j= νj
m,n∈N={, , . . .},
we have the following Hardy-Hilbert-type inequality (cf. [], Theorem ):
∞
m= ∞
n= ambn
Um+Vn
< π sin(π/p)
∞
m= apm
mp–
p∞
n= bqn
nq–
q
. ()
Forμi=νj= (i,j∈N), inequality () reduces to ().
In , Yang and Chen [] gave the following multidimensional Hilbert-type inequality: Fori,j∈N,α,β> ,
xα:=
i
k= x(k)α
α
x=x(), . . . ,x(i)∈Ri,
yβ:=
j
k= y(k)β
β
y=y(), . . . ,y(j)∈Rj,
<λ+η≤i, <λ+η≤j,λ+λ=λ,am,bn≥, we have
n
m
(min{mα,nβ})η
(max{mα,nβ})λ+η
ambn
<K
p
K
q
m
mp(i–λ)–i
α apm
p
n
nq(j–λ)–j
β bqn
q
, ()
wherem=∞mi
=· · ·
∞
m=,
n=
∞
nj=· · · ∞
n=, the series on the right-hand side are
positive, and the best possible constant factorK
p
K
q
is indicated by
K
p
K
q
=
j(
β)
βj–(j
β)
p i(
α)
αi–(i
α)
q λ+ η
(λ+η)(λ+η) .
Fori=j=λ= ,η= ,λ=q,λ=p, inequality () reduces to (). The other results on this type of inequalities were provided by [–].
In , Shi and Yang [] gave another extension of () as follows:
∞
m= ∞
n=
ambn
max{Um,Vn}
<pq ∞
m= apm
mp–
p∞
n= bqn
nq–
q
. ()
Some other results on Hardy-Hilbert-type inequalities were given by [–].
In this paper, by the use of weight coefficients, the transfer formula and the technique of real analysis, a new multidimensional Hilbert-type inequality with multi-parameters and a best possible constant factor is given, which is an extension of () and (). More-over, the equivalent forms, the operator expressions and a few particular inequalities are considered.
2 Some lemmas
Ifμ(ik)> (k= , . . . ,i;i= , . . . ,m),ν(jl)> (l= , . . . ,j;j= , . . . ,n), then we set
Um(k):=
m
i=
μ(ik) (k= , . . . ,i), Vn(l):=
n
j=
νj(l) (l= , . . . ,j), ()
Um=
Um(), . . . ,U(i)
m
, Vn=
Vn(), . . . ,V(j)
n
(m,n∈N).
We also set functionsμk(t) :=μm(k),t∈(m– ,m] (m∈N);νl(t) :=ν(nl),t∈(n– ,n] (n∈N),
Uk(x) :=
x
μk(t)dt (k= , . . . ,i), ()
Vl(y) :=
y
νl(t)dt (l= , . . . ,j), ()
U(x) :=U(x), . . . ,Ui(x)
, V(y) :=V(y), . . . ,Vj(y)
(x,y≥). ()
It follows thatUk(m) =Um(k)(k= , . . . ,i;m∈N),Vl(n) =Vn(l)(l= , . . . ,j;n∈N), and for x∈(m– ,m),Uk(x) =μk(x) =μ(mk)(k= , . . . ,i;m∈N); fory∈(n– ,n),Vl(y) =νl(y) =νn(l)
(l= , . . . ,j;n∈N).
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. ()
Lemma If i∈N,α,M> , (u)is a non-negative measurable function in(, ],and
DM:=
x∈Ri
+;u=
i
i=
xi
M α
≤
, ()
then we have the following transfer formula(cf. []):
· · ·
DM
i
i=
xi
M α
dx· · ·dxs=
Mii(
α)
αi(i
α)
(u)uiα–du. ()
Lemma For i,j∈N,μm(k)≥μ(mk)+(m∈N,k= , . . . ,i),νn(l)≥νn(l+) (n∈N;l= , . . . ,j), α,β> ,ε> ,we have
m
Um–αi–ε
i
k=
μ(mk)≤
i(
α)
εiε/ααi–(i
α)
+O(), ()
n
Vn–j–
ε β
j
k=
νn(k)≤
j(
β)
εjε/ββj–(j
β)
+O()˜ ε→+. ()
Proof ForM>i/α
, we set
(u) = ⎧ ⎨ ⎩
, <u< i
Mα,
(Mu/α)i+ε,
i
Mα ≤u≤.
By (), it follows that
{x∈Ri +;xi≥}
dx
xi+ε α
= lim
M→∞
· · ·
DM
i
i=
xi
M α
dx· · ·dxi
= lim
M→∞
Mii(
α)
αi(i
α)
i/Mα
uiα–
(Mu/α)i+εdu=
i(
α)
εiε/ααi–(i
α)
Then by () and the above result, we find
< {m∈Ni;mi≥}
Um–αi–ε
i
k= μ(mk)
= {m∈Ni;mi≥}
{x∈Ni;m–≤xi<m}
U(m)–i–ε
α
i
k= μ(mk)dx
< {m∈Ni;mi≥}
{x∈Ni;m–≤xi<m}
U(x)–αi–ε
i
k=
μ(mk)(x)dx
=
{x∈Ni;xi≥}
U(x)–αi–ε
i
k=
μ(k)(x)dxν=U=(x)
{ν∈Ri+;νi≥μ(i)}
ν–i–ε
α dν
=
{ν∈Ri+;νi≥}
ν–i–ε
α dν+Oi() =
i(
α)
εiε/ααi–(i
α)
+Oi().
Fori= , <{m∈Ni;mi=}Um–αi–ε
i
k=μ (k)
m <∞; for i≥, μ(i) =maxmμ(mi),b=
i
i=μ(i), in the same way, we find
< {m∈Ni;∃i,mi=}
Um–αi–ε
i
k= μ(mk)
≤ U–αi–ε
i
k= μ(k)+
i
i=
μ(i) {m∈Ni–;mj≥(j=i)}
Um–(αi–)–(ε+)
i
k=(k=i) μ(mk)
=O() +
bi–(
α)
( +ε)(i– )(+ε)/ααi–(i–
α )
+bOi–() <∞.
Then we have
m
Um–αi–ε
i
k=
μ(mk)= {m∈Ni;∃i,mi=}
m
Um–αi–ε
i
k= μ(mk)
+ {m∈Ni;mj≥}
Um–αi–ε
i
k= μ(mk)
≤ i(
α)
εiε/ααi–(i
α)
+O() ε→+.
Hence, we have (). In the same way, we have ().
Definition Forα,β> , <λ+η≤i, <λ+η≤j,λ+λ=λ, we define two weight coefficientsw(λ,n) andW(λ,m) as follows:
w(λ,n) :=
m
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η
Vnλβ
Umi–
λ
α
i
k=
W(λ,m) :=
n
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η
Umλα
Vn j–λ
β
j
l=
νn(l). ()
Example With regard to the assumptions of Definition , we set
kλ(x,y) =
(min{x,y})η
(max{x,y})λ+η (x,y> ).
Then, (i) for fixedy> ,
kλ(x,y)
xi–λ =
⎧ ⎨ ⎩
yλ+ηxi–λ–η, <x<y, yη
xi+λ+η, x≥y,
is decreasing inR+and strictly decreasing in ([y] + ,∞). In the same way, for fixedx> , kλ(x,y)
yj–λ is decreasing inR+and strictly decreasing in ([x] + ,∞). We still have
k(λ) := ∞
kλ(u, )
du u–λ =
∞
(min{u, })η
(max{u, })λ+η
du u–λ
=
uη
u–λdu+
∞
uλ+η
du u–λ =
λ+ η (λ+η)(λ+η)
. ()
(ii) Forb> , we have
d dx
b+xαα =b+xαα–xα–> (x> ).
Hence, form– <xi<m(i= , . . . ,i;m∈N), we haveU(m)α>U(x)αand
(min{U(m)α,Vnβ})η
(max{U(m)α,Vnβ})λ+η
U(m)i–λ
α
< (min{U(x)α,Vnβ})
η
(max{U(x)α,Vnβ})λ+η
U(x)i–λ
α
;
form<xi<m+ (i= , . . . ,i;m∈N), we haveU(m)α<U(x)αand
(min{U(m)α,Vnβ})η
(max{U(m)α,Vnβ})λ+η
U(m)i–λ
α
> (min{U(x)α,Vnβ})
η
(max{U(x)α,Vnβ})λ+η
U(x)i–λ
α
.
Lemma With regard to the assumptions of Definition, (i)we have
w(λ,n) <K(λ) n∈Nj, ()
W(λ,m) <K(λ) m∈Ni, ()
where
K(λ) = j(
β)
βj–(j
β)
k(λ), K(λ) = i(
α)
αi–(i
α)
(ii)forμ(mk)≥μm(k)+ (m∈N),ν (l)
n ≥νn(l+) (n∈N),U (k)
∞ =V∞(l) (k= , . . . ,i,l= , . . . ,j), < λ+η≤i,λ+η> , <ε<pλ(p> ),we have
<K(λ) –θλ(n)
<w(λ,n)
n∈Nj, ()
where,for c:=max≤k≤i{μ
(k) }(> ),
θλ(n) :=
k(λ)
ci/α/Vnβ
(min{v, })ηvλ–
(max{v, })λ+η dv=O
Vnλβ+η
. ()
Proof (i) By (), () and Example (ii), for <λ+η≤i,λ> , it follows that
w(λ,n) =
m
{x∈Ni;mi–≤xi≤mi}
(min{U(m)α,Vnβ})η
(max{U(m)α,Vnβ})λ+η
× Vn
λ
β
U(m)i–λ
α
i
k= μ(mk)dx
<
m
{x∈Ni;mi–≤xi≤mi}
(min{U(x)α,Vnβ})η
(max{U(x)α,Vnβ})λ+η
× Vn
λ
β
U(x)i–λ
α
i
k=
μ(mk)(x)dx
=
Ri +
(min{U(x)α,Vnβ})η
(max{U(x)α,Vnβ})λ+η
Vnλβ
U(x)i–λ
α
i
k=
μ(k)(x)dx
u=U(x)
≤
Ri +
(min{uα,Vnβ})η
(max{uα,Vnβ})λ+η
Vnλβ
ui–λ
α
du
= lim
M→∞
DM
(min{M[i
i=(
ui M)
α]/α,V
nβ})η
(max{M[i
i=(
ui
M)α]/α,Vnβ})λ+η
Mλ–iV
nλβdu
[i
i=(
ui
M)α](i–λ)/α
= lim
M→∞
Mii(
α)
αi(i
α)
(min{Mu/α,V
nβ})ηVnλβ
(max{Mu/α,V
nβ})λ+η
uiα–du
Mi–λu(i–λ)/α
= lim
M→∞
Mii(
α)
αi(i
α)
(min{Mu/α,V
nβ})ηVnλβ
(max{Mu/α,V
nβ})λ+η
uλα–du
v=MuVn/α
β
=
i(
α)
αi–(i
α)
∞
(min{v, })ηvλ–
(max{v, })λ+η dv
=
i(
α)
αi–(i
α)
λ+ η (λ+η)(λ+η)
=K(λ).
Hence, we have (). In the same way, we have (). (ii) By () and in the same way, forc=max≤k≤i{μ
(k)
}(> ), we have
w(λ,n)≥
m
{x∈Ni;mi≤xi≤mi+}
(min{U(m)α,Vnβ})η
(max{U(m)α,Vnβ})λ+η
× Vn
λ
β
U(m)i–λ
α
i
k=
>
m
{x∈Ni;mi≤xi≤mi+}
(min{U(x)α,Vnβ})η
(max{U(x)α,Vnβ})λ+η
× Vn
λ
β
U(x)i–λ
α
i
k=
μ(k)(x)dx
=
[,∞)i
(min{U(x)α,Vnβ})η
(max{U(x)α,Vnβ})λ+η
Vnλβ
U(x)i–λ
α
i
k=
μ(k)(x)dx
v=U(x)
≥
[c,∞)i
(min{vα,Vnβ})η
(max{vα,Vnβ})λ+η
Vnλβ
vi–λ
α
dv.
ForM>ci/α
, we set
(u) = ⎧ ⎨ ⎩
, <u<cαi
Mα,
(min{Mu/α,Vnβ})η
(max{Mu/α,Vnβ})λ+η
Vnλβ
(Mu/α)i–λ, cαi
Mα ≤u≤.
By (), it follows that
{x∈Ri+;xi≥c}
(min{xα,Vnβ})η
(max{xα,Vnβ})λ+η
Vn
λ
β
xi–λ
α
dx
= lim
M→∞
· · · DM
i
i=
xi
M α
dx· · ·dxi
= lim
M→∞
Mii(
α)
αi(i
α)
cαi /Mα
(min{Muα,Vnβ})η
(max{Muα,Vnβ})λ+η
Vnλβu
i α–du
(Muα)i–λ
v=MuVn/α
β
=
i(
α)
αi–(i
α)
ci/α/Vnβ
(min{v, })ηvλ–
(max{v, })λ+η dv.
Hence, we have
w(λ,n) > i(
α)
αi–(i
α)
ci/α/Vnβ
(min{v, })ηvλ–
(max{v, })λ+η dv
=K(λ)
–θλ(n)
> .
ForVnβ≥ci/α, we obtain
<θλ(n) =
k(λ)
ci/α/Vnβ
(min{v, })ηvλ–
(max{v, })λ+η dv
= k(λ)
ci/α/Vnβ
vλ+η–dv= (λ+η)k(λ)
ci/α
Vnβ
λ+η
,
3 Main results
Setting functions
(m) :=Um
p(i–λ)–i
α
(i
k=μ (k)
m)p–
m∈Ni,
(n) :=Vn
q(j–λ)–j
β
(j
l=ν (l)
n)q–
n∈Nj,
and the following normed spaces:
lp,:=
a={am};ap,:=
m
(m)|am|p
p
<∞
,
lq, :=
b={bn};bq, :=
n
(n)|bn|q
q
<∞
,
lp, –p:=
c={cn};cp, –p:=
n
–p(n)|c n|p
p
<∞
,
we have the following.
Theorem If p> , p+q= ,α,β> ,λ> , <λ+η≤i, <λ+η≤j,λ+λ=λ, then for am,bn≥,a={am} ∈lp,,b={bn} ∈lq, ,ap,,bq, > ,we have the following equivalent inequalities:
I:=
n
m
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η
ambn<K
p
(λ)K
q
(λ)ap,bq, , ()
J:=
n
j
k=v (k)
n Vnj–p
λ
β m
(min{Umα,Vnβ})ηam
(max{Umα,Vnβ})λ+η
pp
<K
p
(λ)K
q
(λ)ap,, ()
where
K
p
(λ)K
q
(λ) =
j(
β)
βj–(j
β)
p i(
α)
αi–(i
α)
q
k(λ). ()
Proof By Hölder’s inequality with weight (cf. []), we have
I=
n
m
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η
× Um i–λ
q
α
Vn j–λ
p
β
(j
l=ν (l)
n)
pa m
(i
k=μ (k)
m)
q
Vn j–λ
p
β
Um i–λ
q
α
(i
k=μ (k)
m)
qb n
(j
l=ν (l)
n )
p
≤ m
W(λ,m)Um
p(i–λ)–i
α apm
(i
k=μ (k)
m)p–
p
n
w(λ,n)Vn
q(j–λ)–j
β b
q n
(j
l=ν (l)
n)q–
q
Then by () and (), we have (). We set
bn:=
j
l=ν (l)
n Vnj–p
λ
β m
(min{Umα,Vnβ})ηam
(max{Umα,Vnβ})λ+η
p–
, n∈Nj.
Then we haveJ=bqq–, . Since the right-hand side of () is finite, it followsJ<∞. IfJ= , then () is trivially valid; ifJ> , then by (), we have
bqq, =Jp=I<K
p
(λ)K
q
(λ)ap,bq, ,
bqq–, =J<K
p
(λ)K
q
(λ)ap,,
namely () follows.
On the other hand, assuming that () is valid, by Hölder’s inequality (cf. []), we have
I=
n
(j
l=v (l)
n)/p Vn(βj/p)–λ
m
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η
am
×Vn
(j/p)–λ
β
(j
l=ν (l)
n)/p
bn≤Jbq, . ()
Then by () we have (), which is equivalent to ().
Theorem With regard to the assumptions of Theorem , if μ(mk) ≥μ(mk)+ (m∈ N), νn(l)≥νn(l+) (n∈N),U
(k)
∞ =V∞(l) =∞ (k= , . . . ,i, l= , . . . ,j), then the constant factor
K
p
(λ)K
q
(λ)in()and()is the best possible.
Proof For <ε<p(λ+η),λ˜=λ–pε(∈(–η, –η+i)),λ˜=λ+εp(> –η), we set
˜
a={˜am}, a˜m:=Um–αi+˜λ
i
k=
μ(mk) m∈Ni,
˜
b={˜bn}, b˜n:=Vn–j+
˜
λ–ε
β
j
l=
νn(l) n∈Nj.
Then by () and (), we obtain
˜ap,˜bq, =
m
Ump(i–
λ)–i
α a˜pm
(i
k=μ (k)
m)p–
p
n
Vnq(j–
λ)–j
β b˜
q n
(j
l=ν (l)
n)q–
q
=
m
Um–αi–ε
i
k= μ(mk)
p
n
Vn–j–
ε β
j
l= νn(l)
q
≤
ε
i(
α)
iε/ααi–(i
α)
+εO()
p j(
β)
jε/ββj–(jβ)
+εO()˜
q
By () and (), we find
˜
I:=
n m
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η˜
am
˜
bn
=
n
w(λ˜,n)Vn–j–
ε β
j
l= νn(l)
>K(λ˜)
n
–O
Vn
λ+η
β
Vn–j–
ε β
j
l= νn(l)
=K(λ˜)
j(
β)
εjε/ββj–(j
β)
+O() –˜ O()
.
If there exists a constant K ≤K
p
(λ)K
q
(λ) such that () is valid when replacing
K
p
(λ)K
q
(λ) byK, then we haveεI˜<εK˜ap,˜bq, , namely
K
λ– ε p
j(
β)
jε/ββj–(j
β)
+εO() –˜ εO()
<K
i(
α)
iε/ααi–(i
α)
+εO()
p j(
β)
jε/ββj–(j
β)
+εO()˜
q
.
Forε→+, we find
j(
β)
βj–(j
β)
i(
α)k(λ)
αi–(i
α)
≤K
i(
α)
αi–(i
α)
p j(
β)
βj–(j
β)
q
,
and thenK
p
(λ)K
q
(λ)≤K. Hence,K=K
p
(λ)K
q
(λ) is the best possible constant factor of (). The constant factor 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
With regard to the assumptions of Theorem , in view of
cn:=
j
k=ν (k)
n Vnjβ–pλ m
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η
am
p–
, n∈Nj,
c={cn}, cp, –p=J<K
p
(λ)K
q
(λ)ap,<∞,
we can set the following definition.
Definition Define a multidimensional Hilbert’s operatorT :lp,→lp, –p as follows:
For anya∈lp,, there exists a unique representationTa=c∈lp, –p, satisfying
Ta(n) :=
m
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η
am
Forb∈lq, , we define the following formal inner product ofTaandbas follows:
(Ta,b) :=
n m
(min{Umα,Vnβ})η
(max{Umα,Vnβ})λ+η
am
bn. ()
Then by Theorems and , we have the following equivalent inequalities:
(Ta,b) <K
p
(λ)K
q
(λ)ap,bq, , ()
Tap, –p<K
p
(λ)K
q
(λ)ap,. ()
It follows thatTis bounded with
T:= sup
a(=θ)∈lp,
Tap, –p ap,
≤K
p
(λ)K
q
(λ). ()
Since the constant factorK
p
(λ)K
q
(λ) in () is the best possible, we have
T=K
p
(λ)K
q
(λ) =
j(
β)
βj–(j
β)
p i(
α)
ai–(i
α)
q
k(λ). ()
Remark (i) Forμi=νj= (i,j∈N), () reduces to (). Hence, () is an extension of ().
(ii) Forη= , <λ≤i, <λ≤j, () reduces to the following inequality:
n
m
(max{Umα,Vnβ})λ
ambn
<
j(
β)
βj–(j
β)
p i(
α)
ai–(i
α)
q λ
λλ
ap,bq, . ()
In particular, fori=j=λ= ,λ=q,λ=p, () reduces to (). Hence, () is also an extension of (); so is ().
(iii) Forη= –λ,λ,λ< , () reduces to the following inequality:
n
m
(min{Umα,Vnβ})λ
ambn
<
j(
β)
βj–(j
β)
p i(
α)
ai–(i
α)
q(–λ)
λλap,bq, . () (iv) Forλ= ,λ= –λ(–η<λ<η), () reduces to the following inequality:
n
m
min{Umα,Vnβ}
max{Umα,Vnβ}
η
ambn
<
j(
β)
βj–(j
β)
p i(
α)
ai–(i
α)
q η
η–λ
ap,bq, . ()
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. JZ participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by the National Natural Science Foundation (No. 61370186, No. 61640222), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). We are grateful for their help.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 20 January 2017 Accepted: 23 March 2017
References
1. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1934)
2. Mitrinovi´c, DS, Peˇcari´c, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)
3. Yang, BC: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijin (2009) (in Chinese) 4. Yang, BC, Chen, Q: A multidimensional discrete Hilbert-type inequality. J. Math. Inequal.8(2), 267-277 (2014) 5. Hong, Y: On Hardy-Hilbert integral inequalities with some parameters. J. Inequal. Pure Appl. Math.6(4), Article ID 92
(2005)
6. Zhong, WY, Yang, BC: On multiple Hardy-Hilbert’s integral inequality with kernel. J. Inequal. Appl.2007, Article ID 27962 (2007)
7. Yang, BC, Krni´c, M: On the norm of a multi-dimensional Hilbert-type operator. Sarajevo J. Math.7(20), 223-243 (2011) 8. Krni´c, M, Peˇcari´c, JE, Vukovi´c, P: On some higher-dimensional Hilbert’s and Hardy-Hilbert’s type integral inequalities
with parameters. Math. Inequal. Appl.11, 701-716 (2008)
9. Krni´c, M, Vukovi´c, P: On a multidimensional version of the Hilbert-type inequality. Anal. Math.38, 291-303 (2012) 10. Rassias, M, Yang, BC: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl.
Math. Comput.225, 263-277 (2013)
11. Yang, BC: A multidimensional discrete Hilbert-type inequality. Int. J. Nonlinear Anal. Appl.5(1), 80-88 (2014) 12. Chen, Q, Yang, BC: On a more accurate multidimensional Mulholland-type inequality. J. Inequal. Appl.2014, 322
(2014)
13. Rassias, M, Yang, BC: On a multidimensional Hilbert-type integral inequality associated to the gamma function. Appl. Math. Comput.249, 408-418 (2014)
14. Yang, BC: On a more accurate multidimensional Hilbert-type inequality with parameters. Math. Inequal. Appl.18(2), 429-441 (2015)
15. Huang, ZX, Yang, BC: A multidimensional Hilbert-type integral inequality. J. Inequal. Appl.2015, 151 (2015) 16. Liu, T, Yang, BC, He, LP: On a multidimensional Hilbert-type integral inequality with logarithm function. Math. Inequal.
Appl.18(4), 1219-1234 (2015)
17. Shi, YP, Yang, BC: On a multidimensional Hilbert-type inequality with parameters. J. Inequal. Appl.2015, 371 (2015) 18. Shi, YP, Yang, BC: A new Hardy-Hilbert-type inequality with multi-parameters and a best possible constant factor.
J. Inequal. Appl.2015, 380 (2015)
19. Huang, QL: A new extension of Hardy-Hilbert-type inequality. J. Inequal. Appl.2015, 397 (2015)
20. Wang, AZ, Huang, QL, Yang, BC: A strengthened Mulholland-type inequality with parameters. J. Inequal. Appl.2015, 329 (2015)
21. Yang, BC, Chen, Q: On a Hardy-Hilbert-type inequality with parameters. J. Inequal. Appl.2015, 339 (2015) 22. Li, AH, Yang, BC, He, LP: On a new Hardy-Mulholland-type inequality and its more accurate form. J. Inequal. Appl.
2016, 69 (2016)
23. Rassias, M, Yang, BC: On a Hardy-Hilbert-type inequality with a general homogeneous kernel. Int. J. Nonlinear Anal. Appl.7(1), 249-269 (2016)
24. Chen, Q, Shi, YP, Yang, BC: A relation between two simple Hardy-Mulholland-type inequalities with parameters. J. Inequal. Appl.2016, 75 (2016)
25. Yang, BC, Chen, Q: On a more accurate Hardy-Mulholland-type inequality. J. Inequal. Appl.2016, 82 (2016) 26. Yang, BC: Hilbert-type integral operators: norms and inequalities. In: Pardalos, PM, Georgiev, PG, Srivastava, HM (eds.)