Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 309319, 12 pages
doi:10.1155/2010/309319
Research Article
On a Multiple Hilbert’s Inequality with Parameters
Qiliang Huang
Department of Mathematics, Guangdong Institute of Education, Guangzhou, Guangdong 510303, China
Correspondence should be addressed to Qiliang Huang, qlhuang@yeah.net
Received 12 May 2010; Accepted 31 August 2010
Academic Editor: Wing-Sum Cheung
Copyright q 2010 Qiliang Huang. 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 introducing multiparameters and conjugate exponents and using Hadamard’s inequality
and the way of real analysis, we estimate the weight coefficients and give a multiple more
accurate Hilbert’s inequality, which is an extension of some published results. We also prove
that the constant factor in the new inequality is the best possible and consider its equivalent
form.
1. Introduction
In 1908, Weyl published the following famous Hilbert’s inequality cf. 1. If an , bn ≥ 0, 0 <
∞ 2
∞ 2
n1 an < ∞ and 0 <
n1 bn < ∞, then
∞
∞ am bn
<π
m
n
n1 m1
∞
∞
a2m bn2
m1
n1
1/2
1.1
,
where the constant factor π is the best possible. In 1934, Hardy proved the following more
accurate Hilbert’s inequality cf. 2:
∞ ∞
am bn
<π
mn−1
n1 m1
∞
∞
a2m bn2
m1
n1
where the constant factor π is the best possible. For 0 <
1.1 and 1.2 are given as follows cf. 2:
∞
1/2
n1
,
1.2
a2n < ∞, the equivalent forms of
2
Journal of Inequalities and Applications
2
∞
∞
∞
am
< π 2 a2m ,
mn
n1 m1
m1
∞
∞
am
mn−1
m1
n1
2
< π2
∞
1.3
1.4
a2m ,
m1
where the constant factor π 2 is the best possible. Inequalities 1.1–1.4 are important in
analysis and their applications cf. 3. In near one century, there are many improvements,
generalizations and, applications of 1.1–1.4 in numerous literatures and monographs of
mathematics cf. 2–18. Yang and Huang also considered the multiple Hilbert-type integral
inequality cf. 19, 20. Recently, Yang summarized the methods of introducing parameters
and estimating the weight coefficients to extend Hilbert-type inequalities for the past 100
years. Some representative results are as follows cf. 21, 22:
i if p, r > 1, 1/p 1/q 1/r 1/s 1, 0 < α ≤ 1, 0 < λ ≤ min{r, s}, then
1/p 1/q
∞ ∞
λ λ
1 p1−λ/r−1 p
1 q1−λ/s−1 q
,
×
<B
am
bn
,
m−
n−
λ
r s
2
2
n1 m1 m n − 1
m1
n1
1.5
p
∞ ∞ ∞
am
λ λ p
1 pλ/s−1 1 p1−λ/r−1 p
,
<
B
am ,
1.6
n−
m
−
λ
2
r s
2
n1
m1 m n − 1
m1
∞
∞ am bn
∞
∞ am bn
π
×
α
α <
α
sinπ/r
n−1/2
m−1/2
n1 m1
×
∞ n1
1
n−
2
mn 1
···
n1
pα/s−1 ∞
am
α
α
m1 m − 1/2 n − 1/2
ii if pi , ri > 1,
∞
∞
∞
m1 1
n
n
i1 1/pi 1
i1
mi α
λ
n
i1
n
i
ami
i1 1/ri n−
p
1
2
∞
m1
1
m−
2
q1−α/s−1
p1−α/r−1
1/p
p
am
1.7
1/q
q
bn
,
p
∞
π
1 p1−α/r−1 p
<
am ,
m−
α sinπ/r m1
2
1.8
1, 0 < α ≤ 1, 0 < λα ≤ min1≤i≤n {ri }, then
n
∞
1/pi
α1−n λ
i pi
pi 1−λα/ri −1
ami
<
Γ
mi
.
Γλ i1
ri
mi 1
1.9
The constant factors in the above five inequalities are all the best possible. Inequalities
1.5 and 1.7 are generalizations of inequality 1.2, and inequality 1.9 is a multiple
extension of 1.1. Inequalities 1.6 and 1.8 are the equivalent forms of 1.5 and 1.7,
which are extensions of 1.4.
Journal of Inequalities and Applications
3
In this paper, by introducing multi-parameters and conjugate exponents and using
Hadamard’s inequality, we estimate the weight coefficients and give a multiple more accurate
Hilbert ’s inequality, which is an extension of inequalities 1.5, 1.7, and 1.9. We also prove
that the constant factor in the new inequality is the best possible and consider its equivalent
form.
2. Some Lemmas
Lemma 2.1. If n ∈ N \ {1}, pi , ri > 1 i 1, . . . , n, ni1 1/pi ni1 1/ri 1, λ > 0 < α < 2,
β ≥ −1/2, λα max{1/2 − α, 1} ≤ min1≤i≤n {ri }, then
⎡
⎤1/pi
n
n
λα/ri −11−pi λα/rj −1
⎣ mi β
⎦
mj β
1.
A :
2.1
j1j /
i
i1
Proof. We find the following:
⎡
⎤1/pi
n
n
λα/ri −11−pi 1−λα/ri λα/rj −1
⎣ mi β
⎦
A
mj β
i1
j1
⎡
⎤1/pi
n
n
pi 1−λα/ri λα/rj −1
⎣ mi β
⎦
mj β
i1
2.2
j1
⎡
⎤ni1 1/pi n
1−λα/ri λα/rj −1
⎣
⎦
1,
mi β
mj β
n
i1
j1
and then 2.1 is valid.
Lemma 2.2. If λ, y > 0, r > 1, 1/r 1/s 1, 0 < α < 2, β ≥ −1/2, λα max{1/2 − α, 1} ≤ r,
then
∞ y λ/s m β λα/r−1
1
Γλ/rΓλ/s
Γλ/rΓλ/s
<
1−O
.
α λ <
αΓλ
αΓλ
yλ/r
m1 y m β
2.3
Proof. For fixed y > 0, we set
fx :
λα/r−1
yλ/s x β
α λ ,
y xβ
x ∈ −β, ∞ .
2.4
4
Journal of Inequalities and Applications
In virtue of α λα/r − 2 ≤ 0 and λα/r − 1 ≤ 0, we find −1i f i x > 0, i 1, 2. Putting
u x βα /y, we have the following:
∞
−β
∞
1
α
fxdx 0
uλ/r−1
λ
1 u
du Γλ/rΓλ/s
.
αΓλ
2.5
Since −β ≤ 1/2, by the following Hadamard’s inequality cf. 5:
m1/2
fm <
fxdxm ∈ N,
2.6
m−1/2
it follows that
∞ y λ/s m β λα/r−1
∞
∞ m1/2
fm
<
fxdx
α λ
m1 y m β
m1
m1 m−1/2
∞
∞
Γλ/rΓλ/s
,
fxdx ≤
fxdx αΓλ
1/2
−β
2.7
and then we have the right-hand side of 2.3. Since
1
−β
fxdx xβα /y
α1 uλ
0
1
<
α
uλ/r−1
xβα /y
du
uλ/r−1 du λα/r
r 1β
λαyλ/r
0
2.8
,
and fx is strictly decreasing in −β, ∞, we get
∞
∞
fm >
fxdx ∞
1
m1
>
−β
Γλ/rΓλ/s
−
αΓλ
fxdx −
1
−β
λα/r
r 1β
λαyλ/r
fxdx
2.9
.
Hence, we prove that the left-hand side of 2.3 is valid.
Lemma 2.3. As the assumption of Lemma 2.1, define the weight coefficients ωi mi ωmi ; r1 ,
. . . , rn as
ωi mi : mi β
∞
λα/ri mn 1
···
∞
∞
mi1 1 mi−1 1
···
∞
m1 1
λα/rj −1
mj β
α λ
n i1 mi β
n
j1j /
i
2.10
Journal of Inequalities and Applications
5
i 1, . . . , n, then there exists δn > 0, such that
n
α1−n 1
λ
1−O < ωn mn Γ
δ
Γλ j1
rj
mn β n
mn β
∞
λα/rn ···
mn−1 1
∞
λα/rj −1
n
α1−n λ
j1 mj β
.
Γ
α λ < Γλ
n r
j
β
m
j1
i
i1
n−1 m1 1
2.11
Moreover, for any i ∈ {1, . . . , n}, it follows that
n
λ
α1−n .
Γ
ωi mi <
Γλ j1
rj
2.12
Proof. We prove 2.11 by mathematical induction. For n 2, we set r r1 and s r2
satisfying 1/r 1/s 1. Putting m m1 , y m2 βα , δ2 λα/r > 0, we have the following:
ω2 m2 λα/r1 −1 λα/r2
∞
∞ y λ/s m β λα/r−1
m1 β
m2 β
α α λ
α λ ,
m1 β m2 β
m1 1
m1 y m β
2.13
and then 2.11 is valid by using inequality 2.3.
α
Assuming that for n≥ 2 , 2.11 is valid, then for n 1, setting y n1
i2 mi β >
mn1 βα , s1 1 − 1/r1 −1 , by 2.3, we have the following:
∞ y λ/s1 m β λα/r1 −1
Γλ/r1 Γλ/s1 1
Γλ/r1 Γλ/s1 1
<
.
1 − O1
<
λ/r
λ
α
1
αΓλ
αΓλ
y
y m β
m1 1
2.14
1
Setting λ λ/s1 , rj rj1 /s1 , m
j mj1 j 1, . . . , n, we find nj1 1/rj 1, αλ max{1/2 −
α, 1} ≤ min1≤i≤n {ri }. By the assumption of induction, it follows that
λα/
rj −1
m
j β
ωn1 mn1 m
×
···
n β
α λ
n m
n−1 1
m
1 1
i β
i1 m
⎧
⎫
∞ y λ/s1 m β λα/r1 −1 ⎬
⎨
1
×
⎩m 1 y m βα λ ⎭
1
1
λα/
rn
∞
∞
n−1 j1
λα/
rj −1
j β
Γλ/r1 Γλ/s1 j1 m
< m
n β
···
α λ ·
n αΓλ
m
n−1 1
m
1 1
i β
i1 m
n
n1 Γλ/r1 Γ λ
α1−n α1−n1 ri
λ
< ·
,
Γ
Γ
αΓλ
Γλ
λ
r
i
Γ λ i1
i1
∞
λα/
rn ∞
n−1 2.15
6
Journal of Inequalities and Applications
ωn1 mn1 > m
n β
λα/
rn
∞
×
···
m
n−1 1
∞
m
1 1
λα/
rj −1
m
j β
Γλ/r1 Γλ/s1 α λ ·
n αΓλ
i β
i1 m
n−1 j1
1
yλ/r1
⎡
⎤
λα/
rj −1
n−1 ∞
∞
j β
λα/
rn Γλ/r1 Γλ/s1 ⎢
j1 m
⎥
>
···
n β
⎣ m
α λ − γ ⎦
n αΓλ
m
n−1 1
m
1 1
i β
i1 m
⎛
⎞⎤
⎡
n1 1
α1−n1 λ
⎠⎦ − Γλ/r1 Γλ/s1 γ,
2⎝
>
× ⎣1 − O
Γ
δn
Γλ i1
ri
αΓλ
m
n β
× 1 − O1
2.16
where δn > 0 and
λα/
rj −1
m
j β
1
0 < γ : m
n β
···
αλ/r1
α λ O1 n mn1 β
m
n−1 1
m
1 1
i β
i1 m
n1 1
α1−n ri
× O1 <
Γ
αλ/r1 .
Γλ/s1 i2
λ
mn1 β
rn
λα/
∞
∞
n−1 j1
2.17
Setting δn1 min{δn , αλ/r1 } > 0, by 2.16, we have the following:
n1 λ
1
α1−n1 ,
ωn1 mn1 >
Γ
× 1−O δ
Γλ i1
ri
mn1 β n1
2.18
and then by 2.15, 2.18, and mathematical induction, 2.11 is valid. Setting m
j mj , rj j mj1 , rj rj1 j i, . . . , n − 1, m
n mi , rn ri , then we have the
rj j 1, . . . , i − 1, m
following:
n
n
α1−n λ
λ
α1−n .
Γ
Γ
ωi mi ωm
n ; r1 , . . . , rn <
Γλ j1
Γλ j1
rj
rj
Hence, 2.12 is valid.
2.19
Journal of Inequalities and Applications
7
3. Main Results
Theorem 3.1. Suppose that n ∈ N \ {1}, pi , ri > 1 i 1, . . . , n, ni1 1/pi ni1 1/ri i
1, 1/qn 1 − 1/pn , λ > 0, 0 < α < 2, β ≥ −1/2, λα max{1/2 − α, 1} ≤ min1≤i≤n {ri }, ami ≥
0 mi ∈ N, such that
∞
mi β
0<
pi 1−λα/ri −1 i pi
ami
mi 1
< ∞ i 1, . . . , n,
3.1
then one has the following equivalent inequalities:
I :
∞
∞
···
mn 1
m1 1
1
n i1
mi β
α λ
n
i
ami
i1
1/pi
n
∞
pi 1−λα/ri −1 i pi
α1−n λ
mi β
ami
<
Γ
,
Γλ i1
ri
mi 1
J :
⎧
∞
⎨
⎩m
mn β
λαqn /rn −1
n 1
⎤qn ⎫1/qn
i
⎬
a
i1 mi
⎣
⎦
···
α λ
n ⎭
mn−1 1
m1 1
i1 mi β
⎡
∞
n−1
∞
1/pi
n−1 ∞
pi 1−λα/ri −1 i pi
Γλ/rn λ
ami
< n−1
Γ
.
mi β
ri
α Γλ i1
mi 1
Proof. Since 1/pn 1/qn 1, by 2.1 and Hölder’s inequality cf. 5, we find that
⎡
⎣
∞
···
mn−1 1
⎧
⎪
∞
⎨ ⎪
⎩mn−1 1
n−1
∞
i1
m1 1
···
n ∞
m1 1
i1
i
ami
mi β
n i1
⎤qn
α λ ⎦
1
mi β
⎡
⎤1/pn
n−1
λα/rn −11−pn λα/rj −1
⎦
mj β
α λ ⎣ m n β
⎡
n−1
λα/ri −11−pi × ⎣ mi β
i1
j1
n
j1j / i
mj β
λα/rj −1
3.2
⎤1/pi
⎦
i
ami
⎫qn
⎪
⎬
⎪
⎭
∞
∞
)
p 1−λα/rn −1 *qn /pn 1
≤ ωn mn mn β n
···
α λ
n mn−1 1
m1 1
i1 mi β
⎡
⎤qn /pi
n
n−1
q n
λα/ri −11−pi λα/rj −1
i
⎣ mi β
⎦
ami
×
mj β
i1
j1j /
i
3.3
8
Journal of Inequalities and Applications
≤
n
i1 Γλ/ri αn−1 Γλ
qn /pn
mn β
∞
1−λαqn /rn ···
mn−1 1
∞
m1 1
1
n i1
mi β
α λ
⎡
⎤qn /pi
n
n−1
q n
λα/ri −11−pi λα/rj −1
i
⎣ mi β
⎦
ami ,
×
mj β
i1
j1j /
i
3.4
J≤
n
i1 Γλ/ri αn−1 Γλ
1/pn
⎧
⎪
∞
∞
⎨
⎡
⎤qn /pi
n
n−1
λα/ri −11−pi λα/rj −1
⎣ mi β
⎦
×
···
mjβ
α λ ×
n ⎪
⎩mn 1 mn−1 1 m1 1
i1
j1j /
i
i1 miβ
i
× ami
∞
1
⎫1/qn
⎪
⎬
q n ⎪
⎪
⎪
⎭
⎧
⎪
1/pn ⎪
∞
⎨ i1 Γλ/ri λα/rn −1 ⎤
mn β
⎣
···
α λ ⎦
n ⎪
⎪
β
m
⎩mn−1 1 m1 1 mn 1
i
i1
n
αn−1 Γλ
∞
⎡
∞
⎡
×
⎤qn /pi
n−1 ⎢
i1
⎢ mi β pi 1−λα/ri −1 mi β λα/ri
⎣
n−1
mj β
j1
λα/rj −1 ⎥
⎥
⎦
i
am i
⎫1/qn
⎪
⎬
q n ⎪
j / i
For n ≥ 3, since
following:
J≤
n−1
i1
3.5
⎧
⎪
1/pn n−1 ⎪
∞
⎨ i1 Γλ/ri αn−1 Γλ
λα/rn −1
mn β
···
⎪m 1 m 1 m 1n m βα λ
i1 ⎪
1
n
⎩ n−1
i
i1
∞
∞ ⎡
⎢
pi 1−λα/ri −1 λα/ri
×⎢
mi β
⎣ mi β
n−1
j1
j / i
.
qn /pi 1, by Hölder’s inequality again in 3.5, we have the
n
n
⎪
⎪
⎭
i1 Γλ/ri αn−1 Γλ
1/pn n−1
∞
i1
⎫1/pi
⎤
⎪
⎪
λα/rj −1⎥ i pi ⎬
⎥ am
mj β
i
⎦
⎪
⎪
⎭
ωi mi mi β
mi 1
pi 1−λα/ri −1 i pi
ami
3.6
1/pi
.
Note that for n 2, by 3.5, we directly get 3.6. Hence, 3.3 is valid by 3.6 and 2.12.
Journal of Inequalities and Applications
9
Since 1/qn 1/pn 1, by Hölder’s inequality once again, it follows that
⎡
⎤
n−1 i
∞
∞
∞
+
λα/rn −1/qn 1/qn −λα/rn n ,
ami
i1
⎣ mn β
I
···
amn
α λ ⎦ × mn β
n β
m
mn 1
mn−1 1
m1 1
i
i1
≤J
∞
mn β
pn 1−λα/rn −1
mn 1
n
amn
pn
1/pn
3.7
.
By 3.3, we have 3.2. On the other hand, assuming that 3.2 is valid, setting
n
amn
: mn β
λαqn /rn −1
⎡
⎣
∞
···
mn−1 1
∞
m1 1
n−1
i1
n i1
i
ami
mi β
⎤qn −1
α λ ⎦
,
3.8
then we find that
J
∞
mn β
pn 1−λα/rn −1 n pn
amn
mn 1
1/qn
I 1/qn .
3.9
By 3.2, it follows that J < ∞. If J 0, then 3.3 is naturally valid. Suppose that J > 0, by
3.2, we find that
0<
∞
mn β
pn 1−λα/rn −1 mn 1
n pn
amn
J qn I
1/pi
n
∞
pi 1−λα/ri −1 i pi
i1 Γλ/ri am i
< ∞.
mi β
αn−1 Γλ i1 mi 1
n
<
3.10
Dividing out J qn /pn into two sides of 3.10, we have the following:
∞
mn β
mn 1
pn 1−λα/rn −1 n pn
amn
J
1/pi
n−1
∞
pi 1−λα/ri −1 i pi
i1 Γλ/ri ami
.
mi β
αn−1 Γλ i1 mi 1
n
<
1/qn
3.11
Then 3.3 is valid, which is equivalent to 3.2.
Theorem 3.2. Let the assumptions of Theorem 3.1 be fulfilled, then the same constant factor
α1−n /Γλ ni1 Γλ/ri in 3.2 and 3.3 is the best possible.
10
Journal of Inequalities and Applications
Proof. By 2.11 and
lim mn β
N
λα/rn N →∞
N
···
mn−1 1
λα/rj −1
mj β
α λ ωn mn ,
n i1 mi β
n−1 j1
m1 1
3.12
there exists N0 ∈ N, such that for N > N0 ,
mn β
N
λα/rn N
···
mn−1 1
m1 1
λα/rj −1
n
mj β
1
α1−n λ
1−O ,
Γ
δ
α λ > Γλ
n rj
mn β n
j1
i1 mi β
3.13
n−1 j1
where δn > 0. Setting
i
ami :
⎧
⎨ mi β λα/ri −1 ,
mi ≤ N,
⎩0,
mi > N ,
i 1, . . . , n
3.14
we find that
I :
∞
∞
···
mn 1
m1 1
n 1
i1
mi β
α λ
n
i
ami
i1
λα/rn N
λα/ri −1
n−1 N
N
mn β
i1 mi β
···
α λ
n mn β
m β
mn 1
mn−1 1
m1 1
i
i1
n
N
1
α1−n 1
λ
1−O ·
>
Γ
δ
m β Γλ j1
rj
mn β n
mn 1 n
n
N
1
α1−n λ
Γ
Γλ j1
rj
m β
mn 1 n
×
⎧
⎨
N
1−
1
m
β
mn 1 n
⎩
−1
N
O
mn 1
1
mn β
3.15
⎫
⎬
δn 1
.
⎭
If there exists a constant k ≤ α1−n /Γλ ni1 Γλ/ri , such that 3.2 is still valid as we replace
α1−n /Γλ ni1 Γλ/ri by k, then in particular, we have the following:
I < k
n
∞
mi β
i1
mi 1
pi 1−λα/ri −1 i pi
ami
1/pi
k
N
1
.
m
β
mn 1 n
3.16
Journal of Inequalities and Applications
11
In virtue of 3.15 and 3.16, it follows that
⎧
⎫
−1
n
N
N
⎬
λ ⎨
1
1
α1−n Γ
O 1−
< k.
δ 1
⎭
Γλ j1
rj ⎩
m β
mn β n
mn 1 n
mn 1
3.17
For N → ∞, we have α1−n /Γλ ni1 Γλ/ri ≤ k. Hence, k α1−n /Γλ ni1 Γλ/ri is
the best value of 3.2.
We conform that the constant factor α1−n /Γλ ni1 Γλ/ri in 3.3 is the best
possible, otherwise we can get a contradiction by 3.7 that the constant factor in 3.2 is
not the best possible.
Remarks 3.3. i When 0 < α ≤ 1, the assumption λα max{1/2 − α, 1} ≤ min1≤i≤n {ri } of
two theorems becomes λα ≤ min1≤i≤n {ri }. ii When 0 < α ≤ 1, β 0, 3.2 reduces to
1.9. iii For n 2, r1 r, r2 s, p1 p, p2 q, setting α 1, β −1/2 in 3.2, then
Γλ/r1 Γλ/r2 /Γλ Bλ/r, λ/s, we obtain 1.5. Setting β −1/2, λ 1 in 3.2, we get
1.7.
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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