Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 192197, 13 pages
doi:10.1155/2009/192197
Research Article
On a Multiple Hilbert-Type Integral
Operator and Applications
Qiliang Huang and Bicheng Yang
Department of Mathematics, Guangdong Institute of Education, Guangzhou, Guangdong 510303, China
Correspondence should be addressed to Qiliang Huang, qlhuang@yeah.net
Received 19 August 2009; Accepted 9 December 2009
Academic Editor: Wing-Sum Cheung
Copyright q 2009 Q. Huang and B. Yang. 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.
By using the way of weight functions and the technic of real analysis, a multiple Hilbert-type
integral operator with the homogeneous kernel of −λ-degree λ ∈ R and its norm are considered.
As for applications, two equivalent inequalities with the best constant factors, the reverses, and
some particular norms are obtained.
1. Introduction
∞
1/p
If p > 1, 1/p 1/q 1, f≥ 0 ∈ Lp 0, ∞, g≥ 0 ∈ Lq 0, ∞, fp 0 f p xdx
>
0, gq > 0, then we have the following famous Hardy-Hilbert’s integral inequality and its
equivalent form cf. 1:
∞ ∞
0
0
fxg y
π
dx dy <
f p g q ,
xy
sin π/p
∞ ∞
0
0
fx
dx
xy
p
1/p
dy
<
π
f p ,
sin π/p
1.1
1.2
where the constant factor π/ sinπ/p is the best possible. Define the Hardy-Hilbert’s
∞ integral
p
p
p
operator T : L 0, ∞ → L 0, ∞ as follows: for f ∈ L 0, ∞, T fy : 0 1/x yfxdx y ∈ 0, ∞. Then in view of 1.2, it follows that T fp < π/ sinπ/pfp and
T ≤ π/ sinπ/p. Since the constant factor in 1.2 is the best possible, we find that cf.2
T π/ sinπ/p.
2
Journal of Inequalities and Applications
Inequalities 1.1 and 1.2 are important in analysis and its applications cf. 3. In
2002, reference 4 considered the property of Hardy-Hilbert’s integral operator and gave an
improvement of 1.1 for p q 2. In 2004-2005, introducing another pair of conjugate
exponents r, sr > 1, 1/r 1/s 1 and an independent parameter λ > 0, 5, 6 gave two
best extensions of 1.1 as follows:
∞ ∞
fxg y
π
f g ,
dx dy <
p,φ
q,ψ
λ yλ
λ
sinπ/r
x
0 0
∞ ∞
fxg y
λ λ f g ,
,
dx
dy
<
B
λ
p,φ
q,ψ
r s
0 0
xy
1.3
1.4
where Bu, v is the Beta function φx xp1−λ/r−1 , ψx xq1−λ/s−1 , fp,φ :
∞
1/p
φxf p xdx
> 0, gq,ψ > 0. In 2009, 7, Theorem 9.1.1 gave the following
0
multiple Hilbert-type integral inequality: suppose that n ∈ N \ {1}, pi > 1, ni1 1/pi 1, λ > 0, then kλ x1 , . . . , xn ≥ 0 is a measurable function of −λ-degree in Rn and for any
r1 , . . . , rn ri > 1 satisfies ni1 1/ri 1 and
kλ Rn−1
kλ u1 , . . . , un−1 , 1
n−1
λ/rj −1
uj
du1 · · · dun−1 > 0.
1.5
j1
p
If φi x xpi 1−λ/ri −1 , fi ≥ 0 ∈ Lφii 0, ∞, fpi ,φi > 0i 1, . . . , n, then we have the following
inequality:
Rn
kλ x1 , . . . , xn n
i1
fi xi dx1 · · · dxn < kλ
n fi ,
pi ,φi
1.6
i1
where the constant factor kλ is the best possible. For n 2, kλ x, y 1/xλ yλ , and
λ
1/ x y in 1.6, we obtain 1.3 and 1.4. Inequality 1.6 is some extensions of the results
in 6, 8–11. In 2006, reference 12 also considered a multiple Hilbert-type integral operator
with the homogeneous kernel of −n 1-degree and its inequality with the norm, which is the
best extension of 1.2.
In this paper, by using the way of weight functions and the technic of real analysis,
a new multiple Hilbert-type integral operator with the norm is considered, which is an
extension of the result in 12. As for applications, an extended multiple Hilbert-type integral
inequality and the equivalent form, the reverses, and some particular norms are obtained.
2. Some Lemmas
Lemma 2.1. If n ∈ N \ {1}, pi ∈ R \ {0, 1}, λi ∈ R i 1, . . . , n,
n
i1
1/pi 1, then
⎡
⎤1/pi
n
n
λj −1
λi −11−pi ⎣x
1.
A :
xj ⎦
i
i1
j1j /
i
2.1
Journal of Inequalities and Applications
3
Proof. We find that
A :
n
⎡
⎤1/pi
n
λj −1
⎣xλi −11−pi 1−λi
xj ⎦
i
i1
n
j1
⎡
⎣x1−λi pi
i
i1
n
⎤1/pi
λj −1
xj
⎦
n
j1
⎛
xi1−λi ⎝
n
i1
λj −1
xj
2.2
⎞ni1 1/pi ⎠
,
j1
and then 2.1 is valid.
Definition 2.2. If n ∈ N, Rn : {x1 , . . . , xn xi > 0 i 1, . . . , n}, λ ∈ R, and kλ x1 , . . . , xn is a
measurable function in Rn such that for any u > 0 and x1 , . . . , xn ∈ Rn , kλ ux1 , . . . , uxn u−λ kλ x1 , . . . , xn , then call kλ x1 , . . . , xn the homogeneous function of −λ-degree in Rn .
Lemma 2.3. As for the assumption of Lemma 2.1, if
function of −λ-degree in Rn ,
n
Hi :
Rn−1
kλ u1 , . . . , ui−1 , 1, ui1 , . . . , un i1
λi λ, kλ x1 , . . . , xn ≥ 0 is a homogeneous
n
λj −1
j1j / i
uj
du1 · · · dui−1 dui1 · · · dun
2.3
i 1, . . . , n, and Hn kλ ∈ R, then each Hi Hn kλ i 1, . . . , n, and for any
i 1, . . . , n, it follows that
ωi xi :
xiλi
Rn−1
kλ x1 , . . . , xn n
j1j /
i
λj −1
xj
dx1 · · · dxi−1 dxi1 · · · dxn kλ .
2.4
Proof. Setting uj un vj j / i, n in the integral Hi, we find that
Hi n−1
λj −1
i
kλ v1 , . . . , vi−1 , u−1
dv1 · · · dvi−1 dvi1 · · · dvn−1 dun .
vj u−1−λ
n
n , vi1 , . . . , vn−1 , 1
n−1
R
j1j /
i
2.5
Setting vi u−1
i in
n in the above integral, we obtain Hi Hn. Setting uj xj /xi j /
2.4, we find that ωi xi Hi Hn kλ .
Lemma 2.4. As for the assumption of Lemma 2.3, setting
k λ1 , . . . , λn−1 :
Rn−1
kλ u1 , . . . , un−1 , 1
n−1
j1
λj −1
uj
du1 · · · dun−1 ,
2.6
4
Journal of Inequalities and Applications
then there exist δ0 > 0 and I {λ1 , . . . , λn−1 | λi λi δi , |δi | ≤ δ0 i 1, . . . , n − 1}, such
that for any λ1 , . . . , λn−1 ∈ I, kλ1 , . . . , λn−1 ∈ R, if and only if kλ1 , . . . , λn−1 is continuous at
λ1 , . . . , λn−1 .
Proof. The sufficiency property is obvious. We prove the necessary property of the condition
by mathematical induction. For n 2, since
1
kλ1 δ1 0
kλ u1 , 1uλ1 1 δ1 −1 du1 ∞
1
kλ u1 , 1uλ1 1 δ1 −1 du1 ,
kλ u1 , 1uλ1 1 δ1 −1 ≤ kλ u1 , 1uλ1 1 −δ0 −1 du1 ,
u1 ∈ 0, 1,
kλ u1 , 1uλ1 1 δ1 −1 ≤ kλ u1 , 1uλ1 1 δ0 −1 du1 ,
u1 ∈ 1, ∞,
2.7
and kλ1 − δ0 kλ1 δ0 < ∞, then by Lebesgue control convergence theorem cf. 13, it
follows that kλ1 δ1 kλ1 o1δ1 → 0. Assuming that for n≥ 2, kλ1 , . . . , λn−1 is
continuous at λ1 , . . . , λn−1 , then for n 1, in view of the result for n 2, we have that
lim kλ1 δ1 , . . . , λn δn δn → 0
lim
∞
δn → 0 0
∞
⎛
⎝
⎝
Rn−1
Rn−1
0
⎛
∞
Rn−1
⎞
n−1
λj δj −1
kλ u1 , . . . , un , 1 uj
du1 · · · dun−1 ⎠uλnn δn −1 dun
j1
⎞
n−1
λj δj −1
kλ u1 , . . . , un , 1 uj
du1 · · · dun−1 ⎠uλnn −1 dun
2.8
j1
kλ u1 , . . . , un , 1uλnn −1 dun
n−1
0
λj δj −1
uj
du1 · · · dun−1 ,
j1
then by the assumption for n, it follows that
lim kλ1 δ1 , . . . , λn δn kλ1 , . . . , λn o1
δn → 0
δi −→ 0, i 1, . . . , n − 1.
2.9
By mathematical induction, we prove that for n ∈ N \ {1}, kλ1 , . . . , λn−1 is continuous at
λ1 , . . . , λn−1 .
Lemma 2.5. As for the assumption of Lemma 2.4, if 0 < ε < min1≤i≤n {|pi |}δ0 , then for ε → 0 ,
∞
Iε : ε
1
···
∞
1
kλ x1 , . . . , xn n
j1
λj −ε/pj −1
xj
dx1 · · · dxn kλ o1.
2.10
Journal of Inequalities and Applications
5
Proof. Setting uj xj /xn j 1, . . . , n − 1, we find that
Iε ε
∞
1
⎡
⎤
∞
∞
n−1
λ
−ε/p
−1
j
j
xn−1−ε ⎣
···
kλ u1 , . . . , un−1 , 1 uj
du1 · · · dun−1 ⎦dxn .
xn−1
xn−1
2.11
j1
Setting Dj : {u1 , . . . , un−1 | uj ∈ 0, xn−1 , uk ∈ 0, ∞ k /
j} and
Aj xn :
···
kλ u1 , . . . , un−1 , 1
n−1
Dj
λj −ε/pj −1
uj
du1 · · · dun−1 ,
2.12
j1
then by 2.11, it follows that
Iε ≥
Rn−1
kλ u1 , . . . , un−1 , 1
n−1
λj −ε/pj −1
uj
du1 · · · dun−1 − ε
n−1 ∞
j1
j1
1
xn−1 Aj xn dxn .
2.13
∞
Without loses of generality, we estimate that 1 xn−1 An−1 xn dxn O1. In fact, setting α > 0
such that |ε/pn−1 α| < δ0 , since −uαn−1 ln un−1 → 0 un−1 → 0 , there exists M > 0, such
that −uαn−1 ln un−1 ≤ Mun−1 ∈ 0, 1, and then by Fubini theorem, it follows that
0≤
∞
1
∞
1
xn−1 An−1 xn dxn
1
Rn−2
0
⎡
xn−1 ⎣
1
Rn−2
0
≤M
xn−1
Rn−2
kλ u1 , . . . , un−1 , 1
kλ u1 , . . . , un−1 , 1
n−1
λj −ε/pj −1
uj
u−1
n−1
1
n−1
λj −ε/pj −1
uj
xn−1 dxn
du1 · · · dun−1
− ln un−1 du1 · · · dun−1
j1
Rn−2
Rn−1
· · · dun−2 ⎦dxn
j1
kλ u1 , . . . , un−1 , 1
≤M
⎤
λj −ε/pj −1
uj
dun−1 du1
j1
1
0
kλ u1 , . . . , un−1 , 1
0
n−1
kλ u1 , . . . , un−1 , 1
n−2
λj −ε/pj −1 λn−1 −ε/pn−1 α−1
uj
un−1
du1 · · · dun−1
j1
n−2
λj −ε/pj −1 λn−1 −ε/pn−1 α−1
uj
un−1
du1 · · · dun−1
j1
ε
ε
M · k λ1 − , . . . , λn−2 −
, λn−1 −
p1
pn−2
ε
pn−1
α
< ∞.
2.14
6
Journal of Inequalities and Applications
Hence by 2.13, we have that
Iε ≥
Rn−1
kλ u1 , . . . , un−1 , 1
n−1
λj −ε/pj −1
uj
du1 · · · dun−1 − o1 1.
2.15
j1
By Lemma 2.4, we find that
Iε ≤ ε
∞
1
⎡
∞ ∞
xn−1−ε ⎣
···
kλ u1 , . . . , un−1 , 1
0
×
n−1
0
⎤
λj −ε/pj −1
uj
du1 · · · dun−1 ⎦dxn
2.16
j1
∞
···
∞
0
kλ u1 , . . . , un−1 , 1
n−1
0
λj −ε/pj −1
uj
du1 · · · dun−1
j1
ε
ε
k λ1 − , . . . , λn−1 −
kλ o2 1,
p1
pn−1
Then by combination with 2.15, we have 2.10.
Lemma 2.6. Suppose that n ∈ N \ {1}, p1 ∈ R \ {1}, ni1 1/pi 1, 1/qn 1 − 1/pn , λ1 ,
n
. . . , λn ∈ Rn , i1 λi λ, then kλ x1 , . . . , xn ≥ 0 is a measurable function of −λ-degree in Rn such
that
kλ Rn−1
kλ u1 , . . . , un−1 , 1
n−1
λj −1
uj du1 · · · dun−1 ∈ R.
2.17
j1
If fi ≥ 0 are measurable functions in R i 1, . . . , n − 1, then (1) for pi > 1 i 1, . . . , n,
∞
J :
q λ −1
xnn n
0
≤ kλ
n−1 ∞
i1
Rn−1
kλ x1 , . . . , xn xpi 1−λi −1 f pi xdx
n−1
qn
fi xi dx1 · · · dxn−1
i1
1/pi
,
0
(2) for 0 < p1 < 1, pi < 0 i 2, . . . , n, the reverse of 2.18 is obtained.
1/qn
dxn
2.18
Journal of Inequalities and Applications
7
Proof. 1 For pi > 1 i 1, . . . , n, by Hölder’s inequality cf. 14 and 2.4, it follows that
Rn−1
kλ x1 , . . . , xn n−1
qn
fi xi dx1 · · · dxn−1
i1
⎧
⎪
⎨
⎡
⎤1/pi
n−1
n
λi −11−pi λj −1
kλ x1 , . . . , xn ⎣xi
xj ⎦ fi xi ⎪
⎩ Rn−1
i1
j1j /
i
⎡
n−1
λn −11−pn ×⎣xn
⎤1/pn
λj −1
xj
⎦
dx1 · · · dxn−1
j1
⎫qn
⎪
⎬
⎪
⎭
⎡
⎤qn /pi
n−1
n
λi −11−pi λj −1
≤
kλ x1 , . . . , xn ⎣xi
xj ⎦
Rn−1
i1
j1j /
i
2.19
q
× fi n xi dx1 · · · dxn−1
×
⎧
⎨
⎩
n−1
λn −11−pn Rn−1
kλ x1 , . . . , xn xn
λj −1
xj
dx1 · · · dxn−1
j1
kλ qn −1 xn
1−qn λn
⎫qn −1
⎬
⎭
Rn−1
kλ x1 , . . . , xn ⎡
⎤qn /pi
n−1
n
λi −11−pi λj −1
q
⎣x
×
fi n xi dx1 · · · dxn−1 ,
xj ⎦
i
i1
j1j /
i
J ≤ kλ 1/pn
⎧
⎪
⎨ ∞ ⎪
⎩
Rn−1
0
kλ x1 , . . . , xn ⎡
×
kλ 1/pn
n−1
n
i1
j1j /
i
⎣xλi −11−pi i
⎧
⎪
⎨
⎪
⎩
∞
Rn−1
⎤qn /pi
λj −1
xj
⎦
kλ x1 , . . . , xn xnλn −1 dxn
q
fi n xi dx1 · · · dxn−1 dxn
⎫1/qn
⎪
⎬
⎪
⎭
0
⎫1/qn
⎡
⎤qn /pi
⎪
n−1
n−1
⎬
λi −11−pi λj −1
qn
⎣
⎦
×
fi xi dx1 · · · dxn−1
.
xj
xi
⎪
⎭
i1
j1j /
i
2.20
8
Journal of Inequalities and Applications
For n ≥ 3, by Hölder’s inequality again, it follows that
J ≤ kλ 1/pn
⎧
⎪
n−1 ⎨
⎪
⎩ i1
×
∞
Rn−1
kλ x1 , . . . , xn xnλn −1 dxn
0
λi −11−pi xi
⎤qn /pi ⎫1/qn
⎪
⎬
λj −1 p
xj fi i xi dx1 · · · dxn−1 ⎦
⎪
⎭
j1j /
i
n−1
⎧ ⎡
n−1 ⎨ ∞ ⎣
kλ 1/pn
kλ x1 , . . . , xn ⎩ 0
Rn−1
i1
×
kλ 1/pn
n−1 ∞
i1
0
xiλi
⎫1/pi
⎬
⎤
n
j1j /
i
λj −1
xj dx1
p 1−λ −1 p
· · · dxi−1 dxi1 · · · dxn ⎦xi i i fi i xi dxi
⎭
p 1−λi −1 pi
fi xi dxi
ωi xi xi i
1/pi
.
2.21
Then by 2.4, we have 2.18 note that for n 2, we do not use Hölder’s inequality again.
2 For 0 < p1 < 1, pi < 0 i 2, . . . , n, by the reverse Hölder’s inequality and the same way,
we obtain the reverses of 2.18.
3. Main Results and Applications
p
As for the assumption of Lemma 2.6, setting φi x : x i 1−λi −1 x ∈ 0, ∞; i 1, . . . , n, then
1/1−pn x xqn λn −1 . If pi > 1i 1, . . . , n, then define the following real function
we find that φn
spaces:
p
Lφii 0, ∞
n−1
i1
p
Lφii 0, ∞
:
:
f; f pi,φ ∞
i
)
(p
(
φi x(fx( i dx
f1 , . . . , fn−1 ; fi ∈
T f xn :
p
Lφii 0, ∞, i
Rn−1
<∞
i 1, . . . , n,
0
and a multiple Hilbert-type integral operator T :
+
pi
f f1 , . . . , fn−1 ∈ n−1
i1 Lφi 0, ∞,
1/pi
kλ x1 , . . . , xn n−1
i1
3.1
*
1, . . . , n − 1 ,
+n−1
i1
p
Lφii 0, ∞ → L
fi xi dx1 · · · dxn−1 ,
qn
1/1−pn φn
as follows: for
xn ∈ 0, ∞.
3.2
Journal of Inequalities and Applications
Then by 2.18, it follows that T f ∈ L
and T ≤ kλ , where
9
qn
1/1−pn φn
, T is bounded, T fqn ,φ1/1−pn ≤ kλ
n
T f 1/1−pn qn ,φn
.
sup
T :
+n−1
+n−1 pi
θ,i1,...,n−1
f∈ i1 Lφ 0,∞fi /
i1 fi pi ,φi
+n−1
i1
fi pi ,φi ,
3.3
i
Define the formal inner product of T f1 , . . . , fn−1 and fn as
T f1 , . . . , fn−1 , fn :
Rn
n
kλ x1 , . . . , xn fi xi dx1 · · · dxn .
3.4
i1
Theorem 3.1. Suppose that n ∈ N \ {1}, p1 ∈ R \ {1}, ni1 1/pi 1, 1/qn 1 − 1/pn ,
then kλ x1 , . . . , xn ≥ 0 is a measurable function of −λ-degree in Rn , and for any λ1 , . . . , λn ∈ Rn ,
it satisfies ni1 λi λ and
kλ Rn−1
kλ u1 , . . . , un−1 , 1
n−1
λj −1
uj du1 · · · dun−1 > 0.
3.5
j1
p
If fi ≥ 0 ∈ Lφii 0, ∞, fpi,φ > 0i 1, . . . , n, then (1) for pi > 1 i 1, . . . , n, T kλ and the
i
following equivalent inequalities are obtained:
T f1 , . . . , fn−1 1/1−pn qn ,φn
< kλ
n−1 fi ,
pi ,φi
3.6
i1
n fi ,
T f1 , . . . , fn−1 , fn < kλ
pi ,φi
3.7
i1
where the constant factor kλ is the best possible; (2) for 0 < p1 < 1, pi < 0 i 2, . . . , n, using the
formal symbols of the case in pi > 1 i 1, . . . , n, the equivalent reverses of 3.6 and 3.7 with the
best constant factor are given.
Proof. 1 For pi > 1 i 1, . . . , n, if 2.18 takes the form of equality, then for n ≥ 3 in 2.21,
k such that they are not all zero and
there exist constants Ci and Ck i /
λi −11−pi Ci xi
n−1
j1j / i
λj −1 pi
fi xi xj
n−1
λk −11−pk Ck xk
j1j / k
3.8
λj −1 pk
fk xk xj
p 1−λi pi
p 1−λk pk
fi xi Ck xkk
fk xk C a.e.
p 1−λ −1 p
xi i i fi i xi C/Ci xi , which contradicts fpi,φ
a.e. in Rn ,
in Rn . Assuming that Ci > 0, then
viz. Ci xi i
i
> 0. Note that for n 2, we only
10
Journal of Inequalities and Applications
p
consider 2.19 for fk k xk 1 in the above. Hence we have 3.6. By Hölder’s inequality, it
follows that
T f, fn ∞
λ −1/qn
xnn
0
Rn−1
n−1
kλ x1 , . . . , xn fi xi dx1 · · · dxn−1
i1
1/q −λ
× xn n n fn xn dxn ≤ T f1 , . . . , fn−1 qn ,φ1/1−pn fn pn ,φn ,
3.9
n
and then by 3.6, we have 3.7. Assuming that 3.7 is valid, setting
fn xn :
q λ −1
xnn n
Rn−1
n−1
kλ x1 , . . . , xn fi xi dx1 · · · dxn−1
qn −1
,
3.10
i1
*1/qn
) p 1−λ −1 p
∞
then J 0 xnn n fn n xn dxn
. By 2.18, it follows that J < ∞. If J 0, then 3.6 is
naturally valid. Assuming that 0 < J < ∞, by 3.7, it follows that
∞
p 1−λn −1 pn
fn xn dxn
xnn
0
n fi ,
J qn T f, fn < kλ
pi ,φi
i1
∞
0
p 1−λ −1 p
xnn n fn n xn dxn
1/qn
3.11
n−1 fi ,
J < kλ
pi ,φi
i1
and then 3.6 is valid, which is equivalent to 3.7.
For ε > 0 small enough, setting fi xas follows: fi x 0, x ∈ 0, 1; fi x λi −ε/pi −1
, x ∈ 1, ∞ i 1, . . . , n, if there exists k ≤ kλ , such that 3.7 is still valid as we
x
replace kλ by k, then in particular, by Lemma 2.5, we have that
n kλ o1 Iε ε T f1 , . . . , fn−1 , fn < εk fi i1
pi ,φi
k,
3.12
and kλ ≤ k ε → 0 . Hence k kλ is the best value of 3.7. We conform that the constant
factor kλ in 3.6 is the best possible; otherwise, we can get a contradiction by 3.9 that the
constant factor in 3.7 is not the best possible. Therefore T kλ .
2 For 0 < p1 < 1, pi < 0 i 2, . . . , n, by using the reverse Hölder’s inequality and
the same way, we have the equivalent reverses of 3.6 and 3.7 with the same best constant
factor.
Journal of Inequalities and Applications
11
λ
Example 3.2. For λ > 0, λi λ/ri i 1, . . . , n, ni1 1/ri 1, kλ x1 , . . . , xn 1/ ni1 xi ,
+n
we obtain kλ 1/Γλ i1 Γλ/ri cf. 7, 9.1.19. By Theorem 3.1, it follows that T +
kλ 1/Γλ ni1 Γλ/ri , and then by 3.7, we find that
n
n
R
1
i1 xi
λ
n
Fi xi dx1 · · · dxn
i1
∞
1/pi
n
1 λ
p 1−λ/ri −1 pi
<
Γ
xi i
Fi xi dxi
.
Γλ i1
ri
0
β/n
Setting Fi xi xi fi xi and λi λ/ri −β/ni 1, . . . , n in 3.13, we obtain
λ − β, min1≤i≤n {λi } > −β/n and
,+
n
n
Rn
n
i1 xi
i1
xi
β
λ
3.13
n
i1
λi n
fi xi dx1 · · · dxn
i1
∞
1/pi
n
β
1 p 1−λ −1
<
Γ λi xi i i fi pi xi dxi
.
Γλ i1
n
0
3.14
It is obvious that 3.13 and 3.14 are equivalent in which the constant factors are all the best
β ,+
λ
possible. Hence for kλ−β x1 , . . . , xn n ni1 xi / ni1 xi λ > 0, min1≤i≤n {λi } > −β/n,
+
we can show that T kλ−β 1/Γλ ni1 Γλi β/n.
Example 3.3. For λ > 0, λi λ/ri i 1, . . . , n, ni1 1/ri 1, kλ x1 , . . . , xn +
λ
n
1/max1≤i≤n {xi } , we obtain kλ 1/λn−1 i1 ri cf. 7, 9.1.24. By Theorem 3.1, it follows
+
that T kλ 1/λn−1 ni1 ri , and then by 3.7, we find that
n
1
Rn
Fi xi dx1 · · · dxn
max1≤i≤n {xi }λ i1
∞
1/pi
n
1 p 1−λ/ri −1 pi
< n−1
ri
xi i
Fi xi dxi
.
λ
0
i1
3.15
β/n
Setting Fi xi xi fi xi and λi λ/ri −β/ni 1, . . . , n in 3.15, we obtain
λ − β, min1≤i≤n {λi } > −β/n and
,+
n
i1 xi
n
Rn
β
max1≤i≤n {xi }
<λ
n
i1
n
fi xi dx1 · · · dxn
λ
1
λi β/n
0
p 1−λi −1
xi i
fi pi xi dxi
i1
λi 3.16
i1
∞
n
1/pi
.
12
Journal of Inequalities and Applications
β
,+
Hence for kλ−β x1 , . . . , xn n ni1 xi /max1≤i≤n {xi }λ λ > 0, min1≤i≤n {λi } > −β/n, we
+
can show that T kλ−β λ ni1 1/λi β/n.
Example 3.4. For λ > 0, λi −λ/ri , ni1 1/ri 1 i 1, . . . , n, k−λ x1 , . . . , xn λ
min1≤i≤n {xi } , by mathematical induction, we can show that
k−λ n−1
−λ/rj −1
uj
du1 · · · dun−1 min{u1 , . . . , un−1 , 1}
+n
i1 ri
.
n−1
λ
λ
Rn−1
j1
3.17
In fact, for n 2, we obtain
k−λ 1
0
2 −1
uλ/r
du1
1
∞
1
1 −1
u−λ/r
du1 1
1
r1 r2 .
λ
3.18
Assuming that for n≥ 2 3.17 is valid, then for n 1, it follows that
k−λ n
Rn−1
j2
n
Rn−1
−λ/rj −1
uj
−λ/rj −1
uj
- ∞
0
.
1 −1
du2 · · · dun
du
min{u1 , . . . , un , 1}λ u−λ/r
1
1
min{u ,...,u ,1}
2
n
0
j2
1 −1
uλ1 u−λ/r
du1
1
∞
min{u2 ,...,un ,1}
r12
λr1 − 1
Rn−1
1 −1
du1
min{u2 , . . . , un , 1}λ u−λ/r
1
min{u2 , . . . , un , 1}λ1−1/r1 n
−λ1−1/r1 /1−1/r1 rj −1
uj
du2 · · · dun
3.19
du2 · · · dun
j2
n1
n1 r12
1
1
1
ri .
1
−
r
i
λr1 − 1 λ1 − 1/r1 n−1 i2
r1
λn i1
Then by mathematical induction, 3.17 is valid for n ∈ N \ {1}.
+
By Theorem 3.1, it follows that T k−λ 1/λn−1 ni1 ri , and by 3.7, we find that
Rn
min {xi }
1≤i≤n
λ n
Fi xi dx1 · · · dxn
i1
∞
1/pi
n
1 p 1λ/ri −1 pi
< n−1
ri
xi i
Fi xi dxi
.
λ
0
i1
3.20
Journal of Inequalities and Applications
13
−β/n
Setting Fi xi xi fi xi and λi −λ/ri β/n i 1, . . . , n in 3.20, we obtain
λ
i1 i β − λ, max1≤i≤n {λi } < β/n and
n
n
min1≤i≤n {xi }λ fi xi dx1 · · · dxn
,+
β
Rn
n
n
i1
i1 xi
<λ
n
i1
1
β/n − λi
∞
0
p 1−λi −1
xi i
Hence for kβ−λ x1 , . . . , xn min1≤i≤n {xi }λ /
+
show that T kβ−λ λ ni1 1/β/n − λi .
,+
n
fi pi xi dxi
n
i1 xi
β
3.21
1/pi
.
λ > 0, max1≤i≤n {λi } < β/n, we can
Acknowledgments
This work is supported by the Emphases Natural Science Foundation of Guangdong
Institution, Higher Learning, College and University no. 05Z026, and Guangdong Natural
Science Foundation no. 7004344.
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