Journal of Inequalities in Pure and
Applied Mathematics
ON THE EXTENDED HILBERT’S INTEGRAL INEQUALITY
BICHENG YANG
Department of Mathematics,
Guangdong Institute of Education,
Guangzhou, Guangdong 510303
People’s Republic of China.
volume 5, issue 4, article 96,
2004.
Received 21 May, 2003;
accepted 21 October, 2004.
Communicated by: S.S. Dragomir
EMail: bcyang@pub.guangzhou.gd.cn
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
068-03
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Abstract
This paper gives two distinct generalizations of the extended Hilbert’s integral
inequality with the same best constant factor involving the β function. As applications, we consider some equivalent inequalities.
2000 Mathematics Subject Classification: 26D15.
Key words: Hilbert’s inequality, weight function, β function, Hölder’s inequality.
On the Extended Hilbert’S
Integral Inequality
Research support by the Science Foundation of Professor and Doctor of Guangdong
Institute of Education.
Bicheng Yang
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Results and Applications . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
9
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1.
Introduction
R∞
R∞
If f, g ≥ 0, such that 0 < 0 f 2 (x)dx < ∞ and 0 < 0 g 2 (x)dx < ∞, then
the famous Hilbert’s integral inequality is given by
∞
Z
∞
Z
(1.1)
0
0
f (x)g(y)
dxdy < π
x+y
Z
∞
2
Z
2
f (x)dx
0
12
∞
g (x)dx
,
0
where the constant factor π is the best possible (see [2]). Inequality (1.1) had
been generalized by Hardy-Riesz
R ∞ [1] as:
R∞
If p > 1, p1 + 1q = 1, 0 < 0 f p (x)dx < ∞ and 0 < 0 g q (x)dx < ∞, then
Z
∞
Z
(1.2)
0
0
∞
f (x)g(y)
dxdy
x+y
<
sin
π
π
p
On the Extended Hilbert’S
Integral Inequality
Bicheng Yang
Title Page
Z
∞
p
p1 Z
q
g (x)dx
f (x)dx
0
∞
1q
Contents
,
0
π
where the constant factor sin(π/p)
is the best possible. When p = q = 2, inequality (1.2) reduces to (1.1). We call (1.2) Hardy-Hilbert’s integral inequality,
which is important in analysis and its applications (see [4]).
In recent years, by introducing a parameter λ and the β function, Yang [7, 8]
gave an extension of (1.2) as:R
R∞
∞
If λ > 2−min{p, q}, 0 < 0 x1−λ f p (x)dx < ∞ and 0 < 0 x1−λ g q (x)dx <
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∞, then
Z
∞
Z
(1.3)
0
0
∞
f (x)g(y)
dxdy
(x + y)λ
Z ∞
p1 Z
1−λ p
< kλ (p)
x f (x)dx
0
∞
x
1−λ q
g (x)dx
1q
,
0
p+λ−2
where the constant factor kλ (p) = B p+λ−2
,
is the best possible (B(u, v)
p
p
is the β function). Its equivalent inequality is (see [9, (2.12)]):
p
Z ∞
f (x)
p
(1.4)
y
dx dy < [kλ (p)]
x1−λ f p (x)dx,
λ
(x
+
y)
0
0
0
h ip
where the constant factor [kλ (p)]p = B p+λ−2
is the best possible.
, p+λ−2
p
p
When λ = 1, inequality (1.3) reduces to (1.2), and (1.4) reduces to the equivalent form of (1.2) as:

p
p
Z ∞ Z ∞
Z ∞
f (x)
π 

(1.5)
dx dy <
f p (x)dx.
π
x
+
y
0
0
0
sin p
Z
∞
(λ−1)(p−1)
Z
∞
For p = q = 2, by (1.3), we have λ > 0, and
Z ∞Z ∞
f (x)g(y)
(1.6)
dxdy
(x + y)λ
0
0
Z ∞
12
Z ∞
λ λ
1−λ 2
1−λ 2
<B
,
x f (x)dx
x g (x)dx .
2 2
0
0
On the Extended Hilbert’S
Integral Inequality
Bicheng Yang
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We define (1.6) as the extended Hilbert’s integral inequality. Recently, Yang
et al. [10] provided an extensive account of the above results and Yang [6] gave
a reverse of (1.4) with the same best constant factor. The main objective of
this paper is to build two distinct generalizations of (1.6), with the same best
constant factor but different from (1.3). As applications, we consider some
equivalent inequalities.
For this, we need some lemmas.
On the Extended Hilbert’S
Integral Inequality
Bicheng Yang
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2.
Some Lemmas
We have the formula of the β function as (see [5]):
Z ∞
tu−1
(2.1)
B(u, v) =
dt = B(v, u)
(1 + t)u+v
0
(u, v > 0).
Lemma 2.1 (see [3]). If p > 1, p1 + 1q = 1, ω(σ) > 0, f, g ≥ 0, f ∈ Lpω (E) and
g ∈ Lqω (E) , then the weighted Hölder’s inequality is as follows:
Z
p1 Z
1q
Z
q
p
(2.2)
ω(σ)f (σ)g(σ)dσ ≤
ω(σ)f (σ)dσ
ω(σ)g (σ)dσ ,
E
E
E
On the Extended Hilbert’S
Integral Inequality
where the equality holds if and only if there exists non-negative real numbers A
and B, such that they are not all zero and Af p (σ) = Bg q (σ), a.e. in E.
Bicheng Yang
Lemma 2.2. If r > 1, and λ > 0, define the weight function ωλ (r, x) as
Z ∞
1
λ(1− r1 )
(2.3)
ωλ (r, x) := x
y (λ−r)/r dy.
λ
(x
+
y)
0
Then we have
λ
1
(2.4)
ωλ (r, x) = B
,λ 1 −
.
r
r
Proof. Setting y = xu in the integral of (2.3), we find
Z ∞
(xu)(λ−r)/r
λ(1− r1 )
ωλ (r, x) = x
xdu
xλ (1 + u)λ
0
Z ∞
λ
1
=
u r −1 du.
λ
(1 + u)
0
By (2.1), we have (2.4) and the lemma is proved.
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Note. It is obvious that for p > 1, p1 +
(2.5)
ωλ (p, x) = B
Lemma 2.3. If p > 1, p1 +
I1 :=
∞
= 1 and λ > 0, one has
λ λ
,
p q
= ωλ (q, x).
= 1 and 0 < ε < λ, one has
∞
λ−p−ε
1
p
y
x
dxdy
(x + y)λ
1
1
2
λ−ε λ ε
p
1
> B
, +
−
.
ε
p
q p
λ−ε
Z
(2.6)
1
q
1
q
λ−q−ε
q
Z
Proof. Setting x = yu in I1 , in view of (2.1), one has
Z ∞
Z ∞
λ−p−ε
1
−1−ε
I1 =
y
u p du dy
λ
1
1/y (1 + u)
Z ∞
Z ∞
λ−ε
1
−1
−1−ε
p
=
y
u
du dy
(1 + u)λ
1
0
"Z 1
#
Z ∞
y
λ−ε
1
−
y −1−ε
u p −1 du dy
λ
(1
+
u)
1
0
Z ∞
Z 1
y
λ−ε
λ−ε λ ε
1
−1
> B
, +
−
y
u p −1 dudy.
ε
p
q p
1
0
By calculating the above integral, one has (2.6). The lemma is proved.
On the Extended Hilbert’S
Integral Inequality
Bicheng Yang
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Lemma 2.4. If p > 1, p1 +
I2 :=
(2.7)
= 1 and 0 < ε < λ(p − 1), one has
∞
λ+ε
1
y
xλ− p −1 dxdy
λ
(x + y)
1
1
−2
1
λ ε λ ε
λ ε
> B
− , +
−
−
.
ε
q
p p p
q
p
Z
∞
1
q
−1
λ− λ+ε
q
Z
Proof. Setting x = yu in I2 , in view of (2.1), one has
Z ∞
Z ∞
1
λ− λ+ε
−1
−1−ε
p
I2 =
y
u
du dy
λ
1
1/y (1 + u)
Z ∞
Z ∞
1
λ− λ+ε
−1
−1−ε
=
y
u p du dy
(1 + u)λ
1
0
#
"Z 1
Z ∞
y
λ+ε
1
uλ− p −1 du dy
−
y −1−ε
λ
(1
+
u)
0
1
Z ∞
Z 1
y
λ+ε
λ ε λ ε
1
−1
− , +
−
y
uλ− p −1 dudy.
> B
ε
q
p p p
1
0
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Bicheng Yang
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By calculating the above integral, one has (2.7). The lemma is proved.
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3.
Main Results and Applications
Theorem 3.1. If f, g ≥ 0, p > 1, p1 + 1q = 1, λ > 0, such that 0 <
R∞
< ∞ and 0 < 0 xq−1−λ g q (x)dx < ∞, then
Z
(3.1)
0
∞
Z
R∞
0
xp−1−λ f p (x)dx
∞
f (x)g(y)
dxdy
(x + y)λ
0
1q
Z ∞
p1 Z ∞
λ λ
q−1−λ q
p−1−λ p
x
g (x)dx ;
<B
,
x
f (x)dx
p q
0
0
On the Extended Hilbert’S
Integral Inequality
Bicheng Yang
Z
∞
Z
∞
p
f (x)
dx dy
(x + y)λ
0
0
p Z ∞
λ λ
< B
,
xp−1−λ f p (x)dx,
p q
0
h ip
where the constant factors B λp , λq and B λp , λq
are all the best possible.
Inequality (3.2) is equivalent to (3.1). In particular, for λ = 1, one has the
following two equivalent inequalities:
y λ(p−1)−1
(3.2)
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Z
∞
Z
(3.3)
0
0
∞
f (x)g(y)
dxdy
x+y
Z ∞
p1 Z ∞
1q
π
p−2 p
q−2 q
<
x f (x)dx
x g (x)dx ;
0
0
sin πp
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Z
∞
y
(3.4)
p−2
Z
0
∞
0
f (x)
dx
x+y
p

p
dy < 
sin
π 
π
p
Z
∞
xp−2 f p (x)dx.
0
Proof. By (2.2), one has
Z ∞Z ∞
f (x)g(y)
J1 :=
dxdy
(x + y)λ
0
0
"
# "
#
1
1
Z ∞Z ∞
y q−λ pq
1
xp−λ pq
=
f (x)
g(y) dxdy
(x + y)λ
y q−λ
xp−λ
0
0
(Z "Z
) p1
p−λ 1q #
∞
∞
x
1
≤
dy f p (x)dx
λ
q−λ
(x
+
y)
y
0
0
) 1q
(Z "Z
q−λ p1 #
∞
∞
y
1
(3.5)
dx g q (y)dy
.
×
λ
p−λ
(x + y)
x
0
0
If (3.5) takes the form of an equality, then by Lemma 2.1, there exist real numbers A and B, such that they are not all zero, and
1
A
(x + y)λ
xp−λ
y q−λ
=B
1
q
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f p (x)
1
(x + y)λ
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p1
q−λ
y
xp−λ
g q (y),
a.e. in (0, ∞) × (0, ∞).
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Hence we find
Axp−λ f p (x) = By q−λ g q (y), a.e. in (0, ∞) × (0, ∞).
It follows that there exists a constant C, such that
Axp−λ f p (x) = C,
By
a.e. in (0, ∞);
q−λ q
a.e. in (0, ∞).
g (y) = C,
Without loss of generality, suppose that A 6= 0. One has
xp−λ−1 f p (x) =
C
,
Ax
R∞
On the Extended Hilbert’S
Integral Inequality
a.e. in (0, ∞),
Bicheng Yang
which contradicts the fact that 0 < 0 xp−1−λ f p (x)dx < ∞. Hence, (3.5)
takes the form of strict inequality, and by (2.3), we may rewrite (3.5) as
Z
(3.6)
J1 <
∞
ωλ (q, x)xp−1−λ f p (x)dx
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p1
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0
∞
Z
×
ωλ (p, y)y
q−1−λ q
g (y)dy
0
∼
∼
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∼
f (x) = g(y) = 0,
∼
.
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For 0 < ε < λ, setting f (x) and g(y) as:
f (x) = x
1q
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Hence by (2.5), one has (3.1).
∼
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λ−p−ε
p
∼
x, y ∈ (0, 1);
, g(y) = y
λ−q−ε
q
,
x, y ∈ [1, ∞),
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then we find
Z
(3.7)
∞
x
p−1−λ
p1 Z
∼p
f (x)dx
0
∞
x
q−1−λ ∼q
1q
g (x)dx
0
1
= .
ε
If there exists λ > 0, such that the constant factor in (3.1) is not the
best posλ λ
sible, then there exists a positive number K ( with K < B p , q ), such that
λ λ
(3.1) is still valid if one replaces B p , q by K. In particular, one has
Z
∞
Z
∞
∼
0
On the Extended Hilbert’S
Integral Inequality
∼
f (x)g(y)
εI1 = ε
dxdy
(x + y)λ
0
0
p1 Z
Z ∞
∼p
p−1−λ
x
f (x)dx
< εK
Bicheng Yang
∞
∼q
xq−1−λ g (x)dx
1q
.
0
Hence by (2.6) and (3.7), one has
2
λ−ε λ ε
p
, +
−ε
< K,
B
p
q p
λ−ε
λ λ
and then B p , q ≤ K (ε → 0+ ). This contradicts the fact that K <
B λp , λq . It follows that the constant factor in (3.1) is the best possible.
R∞
Since 0 < 0 xp−1−λ f p (x)dx < ∞, there exists T0 > 0, such that for any
RT
T > T0 , one has 0 < 0 xp−1−λ f p (x)dx < ∞. We set
Z T
p−1
f (x)
λ(p−1)−1
g(y, T ) := y
dx
,
λ
0 (x + y)
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and use (3.1) to obtain
Z T
0<
y q−1−λ g q (y, T )dy
0
Z T
p
Z T
f (x)
λ(p−1)−1
=
y
dx dy
λ
0
0 (x + y)
Z TZ T
f (x)g(y, T )
=
dxdy
(x + y)λ
0
0
1q
Z T
p1 Z T
λ λ
(3.8) < B
,
xp−1−λ f p (x)dx
y q−1−λ g q (y, T )dy .
p q
0
0
Hence we find
Bicheng Yang
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T
Z
y q−1−λ g q (y, T )dy
0<
1− 1q
0
Z
=
T
y λ(p−1)−1
<B
λ λ
,
p q
Z
p p1
f (x)
dx dy
λ
0 (x + y)
p1
xp−1−λ f p (x)dx .
Z
0
(3.9)
On the Extended Hilbert’S
Integral Inequality
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T
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0
R∞
It follows that 0 < 0 y q−1−λ g q (y, ∞)dy < ∞. Hence (3.8) and (3.9) are strict
inequalities as T → ∞. Thus inequality (3.2) holds.
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On the other hand, if (3.2) is valid, by Hölder’s inequality (2.2), one has
Z ∞Z ∞
f (x)g(y)
dxdy
(x + y)λ
0
0
h
Z ∞
Z ∞
i
λ+1−q
f
(x)
− λ+1−q
q
=
y
y q
dx
g(y)
dy
(x + y)λ
0
0
p p1
Z ∞
Z ∞
f (x)
λ(p−1)−1
≤
y
dx dy
(x + y)λ
0
0
Z ∞
1q
q−1−λ q
(3.10)
×
y
g (y)dy .
0
Hence by (3.2), one has (3.1). It follows that (3.2) is equivalent to (3.1).
If the constant factor in (3.2) is not the best possible, one can get a contradiction that the constant factor in (3.1) is not the best possible by using (3.10).
The theorem is thus proved.
Theorem 3.2. If f, g ≥ 0, p > 1,
Z
1
p
+
1
q
= 1, λ > 0, such that
∞
x(p−1)(1−λ) f p (x)dx < ∞
0<
Z
0<
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and
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Integral Inequality
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∞
x
(q−1)(1−λ) q
g (x)dx < ∞,
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then
Z
(3.11)
0
∞
Z
∞
f (x)g(y)
dxdy
(x + y)λ
0
Z ∞
p1
λ λ
<B
,
x(p−1)(1−λ) f p (x)dx
p q
0
Z ∞
1q
(q−1)(1−λ) q
×
x
g (x)dx ;
0
Z
∞
y λ−1
Z
∞
f (x)
dx
(x + y)λ
p
dy
p Z ∞
λ λ
< B
,
x(p−1)(1−λ) f p (x)dx,
p q
0
h ip
where the constant factors B λp , λq and B λp , λq
are the best possible.
Inequality (3.12) is equivalent to (3.11). In particular, for λ = p > 1, one has
the following two equivalent inequalities:
Z ∞Z ∞
f (x)g(y)
(3.13)
dxdy
(x + y)p
0
0
Z ∞ p
p1 Z ∞ q
1q
f (x)
1
g (x)
<
dx
dx
p−1
x
x(p−1)2
0
0
and
Z ∞
p
p Z ∞ p
Z ∞
f (x)
1
f (x)
p−1
(3.14)
y
dx
dy
<
dx.
(x + y)p
p−1
x(p−1)2
0
0
0
(3.12)
0
0
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Integral Inequality
Bicheng Yang
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Proof. By (2.2), one has
Z ∞Z ∞
f (x)g(y)
J1 =
dxdy
(x + y)λ
0
0
"
!
#
Z ∞Z ∞
2
1
x(q−λ)/q
=
f (x)
(x + y)λ
y (p−λ)/p2
0
0
!
#
"
2
y (p−λ)/p
g(y) dxdy
×
x(q−λ)/q2
! #
) p1
(Z "Z
2
∞
∞
x(q−λ)p/q
1
≤
dy f p (x)dx
λ
(p−λ)/p
(x
+
y)
y
0
0
(Z "Z
! #
) 1q
2
∞
∞
1
y (p−λ)q/p
(3.15)
.
×
dx g q (y)dy
λ
(q−λ)/q
(x
+
y)
x
0
0
On the Extended Hilbert’S
Integral Inequality
Bicheng Yang
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Following the same manner as (3.6), one has
Z
(3.16)
J1 <
∞
ωλ (p, x)x
(p−1)(1−λ) p
p1
II
I
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f (x)dx
0
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Z
×
0
∞
ωλ (q, x)x(q−1)(1−λ) g q (x)dx
1q
.
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Page 16 of 19
Hence by (2.5), one has (3.11).
J. Ineq. Pure and Appl. Math. 5(4) Art. 96, 2004
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∼
∼
For 0 < ε < λ(p − 1), setting f (x) and g(y) as:
∼
∼
x, y ∈ (0, 1);
f (x) = g(y) = 0,
∼
f (x) = xλ−1−
λ+ε
p
∼
, g(y) = y λ−1−
λ+ε
q
,
x, y ∈ [1, ∞),
by Lemma 2.4 and the same way of Theorem 3.1, we can show that the constant
factor in (3.11) is the best possible.
In a similar fashion to Theorem 3.1, we can show that (3.12) is valid, which is
equivalent to (3.11). By the equivalence of (3.11) and (3.12), we may conclude
that the constant factor in (3.12) is the best possible. The theorem is proved.
Remark 3.1. (i) For p = q = 2, both inequalities (3.1) and (3.11) reduce
to (1.6). Inequalities (3.1) and (3.11)are distinct
generalizations of (1.6)
λ λ
with the same best constant factor B p , q , but different from (1.3).
(ii) Since inequalities (3.3) and (1.2) are different, we may conclude that inequality (3.1) is not a generalization of (1.3).
(iii) Since all the given inequalities are with the best constant factors, we have
obtained some new results.
On the Extended Hilbert’S
Integral Inequality
Bicheng Yang
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References
[1] G.H. HARDY, Note on a theorem of Hilbert concerning series of positive
terms, Proc. Math. Soc., 23(2) (1925), Records of Proc. XLV-XLVI.
[2] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge University Press, Cambridge, UK, 1952.
[3] JICHANG KUANG, Applied Inequalities, Shangdong Science and Technology Press, Jinan, China, 2003.
[4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Boston, 1991.
[5] ZHUXI WANG AND DUNRIN GUO , An Introduction to Special Functions, Science Press, Beijing, 1979.
[6] BICHENG YANG, A reverse of Hardy-Hilbert’s integral inequality, Journal of Jilin University (Science Edition), 42(4) (2004), 489–493.
[7] BICHENG YANG, On a general Hardy-Hilbert’s inequality with a best
value, Chinese Annals of Math., 21A(4) (2000), 401–408.
On the Extended Hilbert’S
Integral Inequality
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[8] BICHENG YANG, On Hardy-Hilbert’s integral inequality, J. Math. Anal.
Appl., 261 (2001), 295–306.
[9] BICHENG YANG AND L. DEBNATH, On the extended Hardy-Hilbert’s
inequality, J. Math. Anal. Appl., 272 (2002), 187–199.
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[10] BICHENG YANG AND Th.M. RASSIAS, On the way of weight coefficient and research for the Hilbert-type inequality, Math. Inequal. and Applics., 6(4) (2003), 625–658.
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Integral Inequality
Bicheng Yang
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J. Ineq. Pure and Appl. Math. 5(4) Art. 96, 2004
http://jipam.vu.edu.au