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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 130958, 12 pages
doi:10.1155/2009/130958
Research Article
Multidimensional Hilbert-Type Inequalities with
a Homogeneous Kernel
Predrag Vukovi
´
c
Faculty of Teacher Education, University of Zagreb, Savska cesta 77, 10000 Zagreb, Croatia
Correspondence should be addressed to Predrag Vukovi
´
c,
Received 11 July 2009; Revised 10 November 2009; Accepted 18 November 2009
Recommended by Radu Precup
We consider the Hilbert-type inequalities with nonconjugate parameters. The obtaining of the best
possible constants in the case of nonconjugate parameters remains still open. Our generalization
will include a general homogeneous kernel. Also, we obtain the best possible constants in the case
of conjugate parameters when the parameters satisfy appropriate conditions. We also compare our
results with some known results.
Copyright q 2009 Predrag Vukovi
´
c. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
Let 1/p 1/q 1 p>1,f,g≥ 0,
0 <
∞
0
f
p
x
dx < ∞, 0 <
∞
0
g
q
x
dx < ∞.
1.1
The well-known Hardy-Hilbert’s integral inequality see 1 is given by
∞
0
f
x
g
y
x y
dx dy <
π
sin
π/p
∞
0
f
p
xdx
1/p
∞
0
g
q
xdx
1/q
,
1.2
and an equivalent form is given by
∞
0
∞
0
fx
x y
dx
p
dy <
π
sin
π/p
p
∞
0
f
p
x
dx,
1.3
where the constant factors π/sinπ/p and π/sinπ/p
p
are the best possible.
2 Journal of Inequalities and Applications
During the previous decades, the Hilbert-type inequalities were discussed by many
authors, who either reproved them using various techniques or applied and generalized them
in many different ways. For example, we refer to a paper of Yang see 2.Ifn ∈ N \{1} ,p
i
>
1,
n
i1
1/p
i
1,s>0,f
i
≥ 0, satisfy
0 <
∞
0
x
p
i
−s−1
f
p
i
i
x
dx < ∞
i 1, 2, ,n
,
1.4
then
0,∞
n
n
i1
f
i
x
i
n
j1
x
j
s
dx
1
···dx
n
<
1
Γ
s
n
i1
Γ
s
p
i
∞
0
x
p
i
−s−1
f
p
i
i
xdx
1/p
i
, 1.5
where the constant factor 1/Γs
n
i1
Γs/p
i
is the best possible.
Our generalization will include a general homogeneous kernel Kx
1
, ,x
k
: R
n
k
→ R, where k ≥ 2, with k being nonconjugate parameters. The techniques that will be
used in the proofs are mainly based on classical real analysis, especially on the well-known
H
¨
older’s inequality and on Fubini’s theorem. The obtaining of the best possible constants in
the case of nonconjugate parameters seems to be a very difficult problem and it remains still
open.
Let us recall the definition of nonconjugate exponents see 3.Letp and q be real
parameters, such that
p>1,q>1,
1
p
1
q
≥ 1,
1.6
and let p
and q
, respectively, be their conjugate exponents, that is, 1/p 1/p
1and
1/q 1/q
1. Further, define
λ
1
p
1
q
1.7
and note that 0 <λ≤ 1 for all p and q values as in 1.6. In particular, λ 1 holds if and only
if q p
, that is, only when p and q are mutually conjugate. Otherwise, 0 <λ<1, and in such
cases p and q will be referred to as nonconjugate exponents.
Considering p, q,andλ as in 1.6 and 1.7, Hardy et al. 1, proved that there exists a
constant C
p,q
, dependent only on the parameters p and q, such that the following Hilbert-type
inequality holds for all nonnegative functions f ∈ L
p
R
and g ∈ L
q
R
:
∞
0
f
x
g
y
x y
λ
dx dy ≤ C
p,q
f
L
p
R
g
L
q
R
.
1.8
Journal of Inequalities and Applications 3
Conventions
Throughout this paper we suppose that all the functions are nonnegative and measurable, so
that all integrals converge. We also introduce the following notations:
R
n
{
x
x
1
,x
2
, ,x
n
; x
1
,x
2
, ,x
n
> 0
}
,
|
x
|
α
x
1
α
x
2
α
··· x
n
α
1/α
,α>0,
1.9
and let |S
n−1
|
α
2
n
Γ
n
1/α/α
n−1
Γn/α be an area of unit sphere in R
n
in view of α−norm.
2. Main Results
Before presenting our idea and results, we repeat the notion of general nonconjugate
exponents from 3.Letp
i
,i 1, 2, ,k,be the real parameters which satisfy
k
i1
1
p
i
≥ 1,p
i
> 1,i 1, 2, ,k.
2.1
Further, the parameters p
i
, i 1, 2, ,kare defined by the equations
1
p
i
1
p
i
1,i 1, 2, ,k.
2.2
Since p
i
> 1, i 1, 2, ,k, it is obvious that p
i
> 1, i 1, 2, ,k. We define
λ :
1
k − 1
k
i1
1
p
i
.
2.3
It is easy to deduce that 0 <λ≤ 1. Also, we introduce the parameters q
i
, i 1, 2, ,k,
defined by the relations
1
q
i
λ −
1
p
i
,i 1, 2, ,k.
2.4
In order to obtain our results we need to require
q
i
> 0,i 1, 2, ,k. 2.5
It is easy to see that the above conditions do not automatically apply 2.5. Further, it follows
λ
k
i1
1
q
i
,
1
q
i
1 − λ
1
p
i
,i 1, 2, ,k.
2.6
4 Journal of Inequalities and Applications
Of course, if λ 1, then
k
i1
1/p
i
1; so the conditions 2.1–2.4 reduce to the case of
conjugate parameters.
Results in this section will be based on the following general form of Hardy-Hilbert’s
inequality proven in 4. All the measures are assumed to be σ-finite on some Ω measure
space.
Theorem 2.1. Let k, n ∈ N,k≥ 2, and λ, p
i
,p
i
,q
i
,i 1, 2, ,k, be real numbers satisfying 2.1–
2.5.LetK : Ω
k
→ R and φ
ij
: Ω → R, i, j 1, ,k, be nonnegative measurable functions
such that
k
i,j1
φ
ij
x
j
1. Then, for any nonnegative measurable functions f
i
, i 1, 2, ,k,the
following inequalities hold and are equivalent:
Ω
k
K
λ
x
1
, ,x
k
k
i1
f
i
x
i
dμ
1
x
1
···dμ
k
x
k
≤
k
i1
Ω
φ
ii
F
i
f
i
p
i
x
i
dμ
i
x
i
1/p
i
, 2.7
⎛
⎝
Ω
1
φ
kk
F
k
x
k
Ω
k−1
K
λ
x
1
, ,x
k
k−1
i1
f
i
x
i
dμ
1
x
1
···dμ
k−1
x
k−1
p
k
dμ
k
x
k
⎞
⎠
1/p
k
≤
k−1
i1
Ω
φ
ii
F
i
f
i
p
i
x
i
dμ
i
x
i
1/p
i
,
2.8
where
F
i
x
i
⎛
⎝
Ω
k−1
Kx
1
, ,x
k
·
n
j1,j
/
i
φ
q
i
ij
x
j
dμ
1
x
1
···dμ
i−1
x
i−1
dμ
i1
x
i1
···dμ
k
x
k
⎞
⎠
1/q
i
,
i 1, ,k.
2.9
In the same paper the authors discussed the case of equality in inequalities 2.7 and
2.8. They proved that the equality holds in 2.7and analogously in 2.8 if and only if
f
i
x
i
C
i
φ
ii
x
i
q
i
/1−λq
i
F
i
x
i
1−λ
q
i
,C
i
≥ 0,i 1, ,k.
2.10
In the following theorem we give t he most important case where ΩR
n
,the
measures μ
i
,i 1, ,k, are Lebesgue measures, K
α
: 0, ∞
k
→ R is a nonnegative
homogeneous function of degree −s, s > 0, and the functions φ
ij
represent the form φ
ij
x
j
|x
j
|
A
ij
α
where A
ij
∈ R, i, j 1, ,n. In order to obtain the generalizations of some known
results we define
k
α
β
1
, ,β
k−1
:
0,∞
k−1
K
α
1,t
1
, ,t
k−1
t
β
1
1
···t
β
k−1
k−1
dt
1
···dt
k−1
,
2.11
where we suppose that k
α
β
1
, ,β
k−1
< ∞ for β
1
, ,β
k−1
> −1andβ
1
··· β
k−1
k<s 1.
Journal of Inequalities and Applications 5
Due to technical reasons, we introduce real parameters A
ij
,i,j 1, 2, ,ksatisfying
k
i1
A
ij
0,j 1, 2, ,k.
2.12
We also define
α
i
k
j1
A
ij
,i 1, 2, ,k.
2.13
Theorem 2.2. Let k, n ∈ N,k≥ 2, and λ, p
i
,p
i
,q
i
,i 1, 2, ,k, be real numbers satisfying
2.1–2.5.LetK
α
: 0, ∞
k
→ R be nonnegative measurable homogeneous function of degree −s,
s>0, and let A
ij
,i,j 1, ,k, and α
i
,i 1, ,k be real parameters satisfying 2.12 and
2.13.Iff
i
: R
n
→ R, f
i
/
0, i 1, ,kare nonnegative measurable functions, then the following
inequalities hold and are equivalent:
R
n
k
K
λ
α
|
x
1
|
α
, ,
|
x
k
|
α
k
i1
f
i
x
i
dx
1
···dx
k
<L
k
i1
R
n
|
x
i
|
p
i
/q
i
k−1n−sp
i
α
i
α
f
p
i
i
x
i
dx
i
1/p
i
,
R
n
|
x
k
|
−p
k
/q
k
k−1
n−s
−p
k
α
k
α
R
n
k−1
K
λ
α
|
x
1
|
α
, ,
|
x
k
|
α
·
k−1
i1
f
i
x
i
dx
1
···dx
k−1
p
k
dx
k
<L
p
k
k−1
i1
R
n
|
x
i
|
p
i
/q
i
k−1n−sp
i
α
i
α
f
p
i
i
x
i
dx
i
p
k
/p
i
,
2.14
where
L
S
n−1
k−1λ
α
2
k−1nλ
k
α
n − 1 q
1
A
12
, ,n− 1 q
1
A
1k
1/q
1
· k
α
s − k − 1n − 1 − q
2
α
2
− A
22
,n− 1 q
2
A
23
, ,n− 1 q
2
A
2k
1/q
2
···k
α
n − 1 q
k
A
k2
, ,n− 1 q
k
A
k,k−1
,s− k − 1n − 1 − q
k
α
k
− A
kk
1/q
k
,
2.15
q
i
A
ij
> −n, i
/
j and q
i
A
ii
− α
i
> k − 1n − s.
6 Journal of Inequalities and Applications
Proof. Set Kx
1
, ,x
k
K
α
|x
1
|
α
, ,|x
k
|
α
and φ
ij
x
j
|x
j
|
A
ij
in Theorem 2.1, where
k
i1
A
ij
0 for every j 1, ,k. It is enough to calculate the functions F
i
x
i
, i 1, ,k.
By using the n-dimensional spherical coordinates we find
F
q
1
1
x
1
R
n
k−1
K
α
|
x
1
|
α
, ,
|
x
k
|
α
k
j2
x
j
q
1
A
1j
dx
2
···dx
k
S
n−1
k−1
α
2
k−1
n
0,∞
k−1
K
α
|
x
1
|
α
,t
2
, ,t
k
k
j2
t
n−1q
1
A
1j
j
dt
2
···dt
k
.
2.16
Using homogeneity of the function K
α
and the substitutions u
i
t
i
/|x
1
|
α
,i 2, ,k, we
have
F
q
1
1
x
1
S
n−1
k−1
α
2
k−1n
0,∞
k−1
|
x
1
|
−s
α
K
α
1,u
2
, ,u
k
·
k
j2
|
x
1
|
α
u
j
n−1q
1
A
1j
|
x
1
|
k−1
α
du
2
···du
k
S
n−1
k−1
α
2
k−1n
|
x
1
|
k−1n−sq
1
α
1
−A
11
α
k
α
n − 1 q
1
A
12
, ,n− 1 q
1
A
1k
.
2.17
Similarly, by applying the n-dimensional spherical coordinates and homogeneity of the
function K
α
we have
F
q
2
2
x
2
R
n
k−1
K
α
|
x
1
|
α
, ,
|
x
k
|
α
k
j1,j
/
2
x
j
q
2
A
2j
dx
1
dx
3
···dx
k
S
n−1
k−1
α
2
k−1
n
0,∞
k−1
t
−s
1
K
α
1,
|
x
2
|
α
t
1
,
t
3
t
1
, ,
t
k
t
1
·
k
j1,j
/
2
t
n−1q
2
A
2j
j
dt
1
dt
3
···dt
k
.
2.18
Using the change of variables
t
1
|
x
2
|
α
u
−1
2
,t
i
|
x
2
|
α
u
−1
2
u
i
,i 3, ,k, so
∂
t
1
,t
3
, ,t
k
∂
u
2
,u
3
, ,u
k
|
x
2
|
k−1
α
u
−k
2
,
2.19
Journal of Inequalities and Applications 7
where ∂t
1
,t
3
, ,t
k
/∂u
2
,u
3
, ,u
k
denotes the Jacobian of the transformation, we have
F
q
2
2
x
2
S
n−1
k−1
α
2
k−1n
|
x
2
|
k−1n−sq
2
α
2
−A
22
α
·
0,∞
k−1
K
α
1,u
2
, ,u
k
u
s−k−1n−q
2
α
2
−A
22
2
k
j3
u
n−1q
2
A
2j
j
du
2
···du
k
S
n−1
k−1
α
2
k−1n
|
x
2
|
k−1n−s−q
2
α
2
−A
22
α
· k
α
s −
k − 1
n − 1 − q
2
α
2
− A
22
,n− 1 q
2
A
23
, ,n− 1 q
2
A
2k
.
2.20
In a similar manner we obtain
F
q
i
i
x
i
S
n−1
k−1
α
2
k−1n
|
x
i
|
k−1n−sq
i
α
i
−A
ii
α
· k
α
n − 1 q
i
A
i2
, ,n− 1 q
i
A
i,i−1
,s−
k − 1
n − 1 − q
i
α
i
− A
ii
,
n−1 q
i
A
i,i1
, ,n− 1 q
i
A
ik
2.21
for i 3, ,k. This gives inequalities 2.14 with inequality sign ≤. Condition 2.10
immediately gives that nontrivial case of equality in 2.14 leads to the divergent integrals.
This completes the proof.
Remark 2.3. Note that the kernel K
α
|x
1
|
α
, ,|x
k
|
α
k
i1
|x
i
|
β
α
−s
is a homogeneous
function of degree −βs. In this case we have
k
α
β
1
, ,β
k−1
0,∞
k−1
k−1
i1
t
β
i
i
1
k−1
i1
t
β
i
i
s
dt
1
···dt
k−1
1
β
k−1
Γ
s
Γ
s −
k−1
i1
β
i
1
β
k−1
i1
Γ
β
i
1
β
,
2.22
where we used the well-known formula f or gamma function see, e.g., 5, Lemma 5.1.Now,
by using Theorem 2.2 and 2.22 we obtain the result of Krni
´
cetal.see 6.
3. The Best Possible Constants in the Conjugate Case
In this section we consider the inequalities in Theorem 2.2. In such a way we shall obtain the
best possible constants for some general cases.
It follows easily that Theorem 2.2 in the conjugate case λ 1,p
i
q
i
becomes as
follows.
8 Journal of Inequalities and Applications
Theorem 3.1. Let k, n ∈ N,k≥ 2 and let p
1
, ,p
k
be conjugate parameters such that p
i
> 1,i
1, ,k. Let K
α
: 0, ∞
k
→ R be nonnegative measurable homogeneous function of degree −s,
s>0, and let A
ij
,i,j 1, ,k,and α
i
,i 1, ,kbe real parameters satisfying 2.12 and 2.13.
If f
i
: R
n
→ R, f
i
/
0, i 1, ,k are nonnegative measurable functions, then the following
inequalities hold and are equivalent:
R
n
k
K
α
|
x
1
|
α
, ,
|
x
k
|
α
k
i1
f
i
x
i
dx
1
···dx
k
<M
k
i1
R
n
|
x
i
|
k−1n−sp
i
α
i
α
f
p
i
i
x
i
dx
i
1/p
i
,
R
n
|
x
k
|
1−p
k
k−1n−s−p
k
α
k
α
R
n
k−1
K
α
|
x
1
|
α
, ,
|
x
k
|
α
·
k−1
i1
f
i
x
i
dx
1
···dx
k−1
p
k
dx
k
<M
p
k
k−1
i1
R
n
|
x
i
|
k−1n−sp
i
α
i
α
f
p
i
i
x
i
dx
i
p
k
/p
i
,
3.1
where
M
S
n−1
k−1
α
2
k−1n
k
α
n − 1 p
1
A
12
, ,n− 1 p
1
A
1k
1/p
1
· k
α
s − k − 1n − 1 − p
2
α
2
− A
22
,n− 1 p
2
A
23
, ,n− 1 p
2
A
2k
1/p
2
···k
α
n − 1 p
k
A
k2
, ,n− 1 p
k
A
k,k−1
,s− k − 1n − 1 − p
k
α
k
− A
kk
1/p
k
,
3.2
p
i
A
ij
> −n, i
/
j and p
i
A
ii
− α
i
> k − 1n − s.
To obtain a case of the best inequality it is natural to impose the following conditions
on the parameters A
ij
:
n p
j
A
ji
s −
k − 1
n − p
i
α
i
− A
ii
,j
/
i, i, j ∈
{
1, 2, ,k
}
. 3.3
In that case the constant M from Theorem 3.1 is simplified to the following form:
M
∗
S
n−1
k−1
α
2
k−1
n
k
α
n − 1
A
2
, ,n− 1
A
k
,
3.4
where
A
i
p
1
A
1i
for i
/
1,
A
1
p
k
A
k1
.
3.5
Journal of Inequalities and Applications 9
Further, by using 3.4 and 3.5, the inequalities 3.1 with the parameters A
ij
, satisfying the
relation 3.3, become
R
n
k
K
α
|
x
1
|
α
, ,
|
x
k
|
α
k
i1
f
i
x
i
dx
1
···dx
k
<M
∗
k
i1
R
n
|
x
i
|
−n−p
i
A
i
α
f
p
i
i
x
i
dx
i
1/p
i
, 3.6
⎡
⎣
R
n
|
x
k
|
1−p
k
−n−p
k
A
k
α
R
n
k−1
K
α
|
x
1
|
α
, ,
|
x
k
|
α
·
k−1
i1
f
i
x
i
dx
1
···dx
k−1
p
k
dx
k
⎤
⎦
1/p
k
<M
∗
k−1
i1
R
n
|
x
i
|
−n−p
i
A
i
α
f
p
i
i
x
i
dx
i
1/p
i
.
3.7
Theorem 3.2. Suppose that the real parameters A
ij
,i,j 1, ,ksatisfy conditions in Theorem 3.1
and conditions given in 3.3. If the kernel K
α
t
1
, ,t
k
is as in Theorem 3.1 and for every i 2, ,k
K
α
1,t
2
, ,t
i
, ,t
k
≤ CK
α
1,t
2
, ,0, ,t
k
, 0 ≤ t
i
≤ 1,t
j
≥ 0,j
/
i 3.8
for some C>0, then the constant M
∗
is the best possible in inequalities 3.6 and 3.7.
Proof. Let us suppose that the constant factor M
∗
given by 3.4 is not the best possible in the
inequality 3.6. Then, there exists a positive constant M
1
<M
∗
, such that 3.6 is still valid
when we replace M
∗
by M
1
.
We define the real functions
f
i,ε
: R
n
→ R by the formulas
f
i,ε
x
i
⎧
⎨
⎩
0,
|
x
i
|
α
< 1,
|
x
i
|
α
A
i
−ε/p
i
,
|
x
i
|
α
≥ 1,
i 1, ,k, 3.9
where 0 <ε<min
1≤i≤k
{p
i
p
i
A
i
}. Now, we shall put these functions in inequality 3.6.
By using the n-dimensional spherical coordinates, the right-hand side of the inequality 3.6
becomes
M
1
k
i1
|
x
i
|
α
≥1
|
x
i
|
−n−ε
α
dx
i
1/p
i
M
1
S
n−1
α
2
n
∞
1
t
−1−ε
dt
M
1
S
n−1
α
2
n
ε
.
3.10
10 Journal of Inequalities and Applications
Further, let J denotes the left-hand side of the inequality 3.6, for the above choice of
the functions
f
i,ε
. By applying the n-dimensional spherical coordinates and the substitutions
u
i
t
i
/t
1
,i
/
2, we find
J
|
x
1
|
α
≥1
···
|
x
k
|
α
≥1
K
α
|
x
1
|
α
, ,
|
x
k
|
α
k
i1
|
x
i
|
A
i
−ε/p
i
α
dx
1
···dx
k
S
n−1
k
α
2
kn
∞
1
···
∞
1
K
α
t
1
, ,t
k
k
i1
t
n−1
A
i
−ε/p
i
i
dt
1
···dt
k
S
n−1
k
α
2
kn
∞
1
t
−1−ε/β
1
∞
1/t
1
···
∞
1/t
1
K
α
1,u
2
, ,u
k
k
i2
u
n−1
A
i
−ε/p
i
i
du
2
···du
k
dt
1
.
3.11
Now, it is easy to see that the following inequality holds:
J ≥
S
n−1
k
α
2
kn
∞
1
t
−1−ε
1
∞
0
···
∞
0
K
α
1,u
2
, ,u
k
k
i2
u
n−1
A
i
−ε/p
i
i
du
2
···du
k
dt
1
−
S
n−1
k
α
2
kn
∞
1
t
−1−ε
1
k
j2
I
j
t
1
dt
1
,
3.12
where for j 2, ,k, I
j
t
1
is defined by
I
j
t
1
D
j
K
α
1,u
2
, ,u
k
k
i2
u
n−1
A
i
−ε/p
i
i
du
2
···du
k
,
3.13
satisfying D
j
{u
2
, ,u
k
;0<u
j
< 1/t
1
, 0 <u
l
< ∞,l
/
j}. Without losing generality, we
only estimate the integral I
2
t
1
. For k 2 w e have
I
2
t
1
1/t
1
0
K
α
1,u
2
u
n−1
A
2
−ε/p
2
2
du
2
≤ C
1/t
1
0
u
n−1
A
2
−ε/p
2
2
du
2
C
n
A
2
−
ε
p
2
−1
t
ε/p
2
−n−
A
2
1
,
3.14
Journal of Inequalities and Applications 11
and for k>2wefind
I
2
t
1
≤ C
0,∞
k−2
K
α
1, 0,u
3
, ,u
k
k
i3
u
i
n−1
A
i
−ε/p
i
du
3
···du
k
1/t
1
0
u
2
n−1
A
2
−ε/p
2
du
2
C
n −
ε
p
2
A
2
−1
t
1
ε/p
2
−
A
2
−n
k
α
n − 1
A
3
−
ε
p
3
, ,n− 1
A
k
−
ε
p
k
,
3.15
where k
α
n − 1
A
3
− ε/p
3
, ,n− 1
A
k
− ε/p
k
is well defined since obviously
A
3
···
A
k
<s− k − 2n. Hence, we have I
j
t
1
≤ t
ε/p
j
−n−
A
j
1
O
j
1, for ε → 0
,j∈{2, ,k}, and
consequently
∞
1
t
−1−ε
1
k
j2
I
j
t
1
dt
1
≤ O
1
.
3.16
We conclude, by using 3.10, 3.12,and3.16,thatM
∗
≤ M
1
which is an obvious
contradiction. It follows that the constant M
∗
in 3.6 is the best possible.
Finally, the equivalence of the inequalities 3.6 and 3.7 means that the constant M
∗
is also the best possible in the inequality 3.7. That completes the proof.
Remark 3.3. If we put k 2,K
α
x, yln|x|
α
/|y|
α
/|x|
s
α
−|y|
s
α
,
A
1
s/q − n and
A
2
s/p − n in the inequalities 3.6 and 3.7 applying Theorem 3.2, we obtain the result of Baoju
Sun see 7. Further, by putting n 1 in Theorems 3.1 and 3.2 we obtain appropriate results
from 8. More precisely, the inequality 3.6 becomes
0,∞
k
K
α
x
1
, ,x
k
k
i1
f
i
x
i
dx
1
···dx
k
<M
∗
k
i1
∞
0
x
−1−p
i
A
i
i
f
p
i
i
x
i
dx
i
1/p
i
.
3.17
If the kernel K
α
x
1
, ,x
k
and the parameters A
ij
satisfy the conditions from Theorem 3.2,
then the constant M
∗
k
α
A
2
, ,
A
k
is the best possible. For example, setting
K
α
x
1
, ,x
k
x
1
··· x
k
−s
,s>0,
A
i
s − p
i
/p
i
,i 2, ,k, in the inequality
3.17, we obtain Yang’s result 1.5 from introduction.
Acknowledgment
This research is supported by the Croatian Ministry of Science, Education and Sports, Grant
no. 058-1170889-1050.
12 Journal of Inequalities and Applications
References
1 G. H. Hardy, J. E. Littlewood, and G. P
´
olya, Inequalities, Cambridge University Press, Cambridge, UK,
2nd edition, 1967.
2 B. Yang, “On a new multiple extension of Hilbert’s integral inequality,” Journal of Inequalities in Pure
and Applied Mathematics, vol. 6, no. 2, article 39, pp. 1–8, 2005.
3 F. F. Bonsall, “Inequalities with non-conjugate parameters,” The Quarterly Journal of Mathematics, vol. 2,
pp. 135–150, 1951.
4 I. Brneti
´
c, M. Krni
´
c, and J. Pe
ˇ
cari
´
c, “Multiple Hilbert and Hardy-Hilbert inequalities with non-
conjugate parameters,” Bulletin of the Australian Mathematical Society, vol. 71, no. 3, pp. 447–457, 2005.
5 B. Yang and Th. M. Rassias, “On the way of weight coefficient and research for the Hilbert-type
inequalities,” Mathematical Inequalities & Applications, vol. 6, no. 4, pp. 625–658, 2003.
6 M. Krni
´
c, J. Pe
ˇ
cari
´
c, and P. Vukovi
´
c, “On some higher-dimensional Hilbert’s and Hardy-Hilbert’s
integral inequalities with parameters,” Mathematical Inequalities & Applications, vol. 11, no. 4, pp. 701–
716, 2008.
7 B. Sun, “A multiple Hilbert-type integral inequality with the best constant factor,” Journal of Inequalities
and Applications, vol. 2007, Article ID 71049, 14 pages, 2007.
8 I. Peri
´
candP.Vukovi
´
c, “Hardy-Hilbert’s inequality with general homogeneous kernel,” Mathematical
Inequalities & Applications, vol. 12, no. 3, pp. 525–536, 2009.
