Journal of Inequalities in Pure and
Applied Mathematics
ON A NEW MULTIPLE EXTENSION OF HILBERT’S
INTEGRAL INEQUALITY
volume 6, issue 2, article 39,
2005.
BICHENG YANG
Department of Mathematics,
Guangdong Institute of Education,
Guangzhou, Guangdong 510303,
People’s Republic Of China.
Received 04 June, 2004;
accepted 18 March, 2005.
Communicated by: B.G. Pachpatte
EMail: bcyang@pub.guangzhou.gd.cn
URL: http://page.gdei.edu.cn/~yangbicheng
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
114-04
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Abstract
This paper gives a new multiple extension of Hilbert’s integral inequality with a
best constant factor, by introducing a parameter λ and the Γ function. Some
particular results are obtained.
2000 Mathematics Subject Classification: 26D15.
Key words: Hilbert’s integral inequality; Weight coefficient , Γ function.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Proof of the Theorem and Remarks . . . . . . . . . . . . . . . . . . . . . . 13
References
On a New Multiple Extension of
Hilbert’s Integral Inequality
Bicheng Yang
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1.
Introduction
If f, g ≥ 0 satisfy
Z
0<
∞
∞
Z
2
f (x)dx < ∞
g 2 (x)dx < ∞,
and 0 <
0
0
then
∞
Z
∞
Z
(1.1)
0
0
f (x)g(y)
dxdy < π
x+y
Z
∞
Z
2
2
f (x)dx
0
12
∞
g (x)dx
On a New Multiple Extension of
Hilbert’s Integral Inequality
,
0
where the constant factor π is the best possible (cf. Hardy et al. [2]). Inequality
(1.1) is well known as Hilbert’s integral inequality, which had been extended by
Hardy [1] as:
If p > 1, p1 + 1q = 1, f, g ≥ 0 satisfy
Z
∞
Z
p
f (x)dx < ∞
0<
0
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∞
and 0 <
Bicheng Yang
g q (x)dx < ∞,
0
II
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then
Z
∞
Z
(1.2)
0
0
∞
Close
f (x)g(y)
dxdy
x+y
<
sin
π
π
p
Quit
Z
∞
p
p1 Z
f (x)dx
0
∞
q
g (x)dx
1q
Page 3 of 18
,
0
J. Ineq. Pure and Appl. Math. 6(2) Art. 39, 2005
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π
where the constant factor sin(π/p)
is the best possible. Inequality (1.2) is called
Hardy- Hilbert’s integral inequality, and is important in analysis and its applications (cf. Mitrinović et al.[6]).
Recently, by introducing a parameter λ, Yang [9] gave an extension of (1.2)
as:
If λ > 2 − min{p, q}, f, g ≥ 0 satisfy
Z ∞
Z ∞
1−λ p
0<
x f (x)dx < ∞ and 0 <
x1−λ g q (x)dx < ∞,
0
On a New Multiple Extension of
Hilbert’s Integral Inequality
0
then
Bicheng Yang
Z
∞Z
∞
(1.3)
0
0
f (x)g(y)
dxdy
(x + y)λ
Z
< kλ (p)
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∞
x1−λ f p (x)dx
p1 Z
∞
x1−λ g q (x)dx
1q
,
0
0
where the constant factor kλ (p) = B p+λ−2
, q+λ−2
is the best possible (B(u, v)
p
q
is the β function). For λ = 1, inequality (1.3) reduces to (1.2).
On the problem for multiple extension of (1.1), [3, 4] gave some new results
and Yang [8] gave an improvement of their works as:
P
If n ∈ N\{1}, pi > 1, ni=1 p1i = 1, λ > n − min {pi }, fi ≥ 0, satisfy
1≤i≤n
Z
0<
0
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∞
xn−1−λ fipi (x)dx < ∞
(i = 1, 2, . . . , n),
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then
Z
∞
Z
∞
···
(1.4)
0
P
n
j=1
0
n
Y
1
xj
λ
fi (xi )dx1 . . . dxn
i=1
Z ∞
p1
n
i
1 Y
pi + λ − n
n−1−λ pi
<
x
fi (x)dx
Γ
,
Γ(λ) i=1
pi
0
Qn
pi +λ−n
1
where the constant factor Γ(λ)
Γ
is the best possible. For n = 2,
i=1
pi
inequality (1.4) reduces to (1.3). It follows that (1.4) is a multiple extension of
(1.3), (1.2) and (1.1).
In 2003, Yang et. al [11] provided an extensive account of the above results.
The main objective of this paper is to build a new extension of (1.1) with a
best constant factor other than (1.4), and give some new particular results. That
is
P
Theorem 1.1. If n ∈ N\{1}, pi > 1, ni=1 p1i = 1, λ > 0, fi ≥ 0, satisfy
Z ∞
0<
xpi −1−λ fipi (x)dx < ∞ (i = 1, 2, . . . , n),
0
∞
Z
···
(1.5)
0
0
Bicheng Yang
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then
Z
On a New Multiple Extension of
Hilbert’s Integral Inequality
∞
n
Y
1
P
n
j=1
xj
λ
Close
fi (xi )dx1 . . . dxn
Quit
i=1
Page 5 of 18
Z ∞
p1
i
1
λ
pi −1−λ pi
<
Γ
x
fi (x)dx
,
Γ(λ) i=1
pi
0
n
Y
J. Ineq. Pure and Appl. Math. 6(2) Art. 39, 2005
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where the constant factor
1
Γ(λ)
Qn
i=1 Γ
λ
pi
is the best possible. In particular,
(a) for λ = 1, we have
Z
∞
Z
···
(1.6)
0
0
∞
Qn
i=1 fi (xi )
P
dx1 . . . dxn
n
j=1 xj
p1
Z ∞
n
Y
i
1
pi −2 pi
<
Γ
x
fi (x)dx
;
pi
0
i=1
(b) for n = 2, using the symbol of (1.3) and setting e
kλ (p) = B λp , λq , we
have
Z ∞Z ∞
f (x)g(y)
(1.7)
dxdy
(x + y)λ
0
0
Z ∞
p1 Z ∞
1q
p−1−λ
p
q−1−λ
q
<e
kλ (p)
x
f (x)dx
x
g (x)dx ,
0
0
where the constant factors in (1.6) and (1.7) are still the best possible.
On a New Multiple Extension of
Hilbert’s Integral Inequality
Bicheng Yang
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In order to prove the theorem, we introduce some lemmas.
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2.
Some Lemmas
Lemma 2.1. If k ∈ N, ri > 1 (i = 1, 2, . . . , k + 1), and
∞
Z
∞
Z
···
(2.1)
0
0
k
Y
1
1+
Pk
j=1
uj
λ(k)
r −1
uj j du1
Pk+1
i=1
ri = λ(k), then
Qk+1
. . . duk =
j=1
Γ(ri )
.
Γ(λ(k))
i=1
Proof. We establish (2.1) by mathematical induction. For k = 1, since r1 +r2 =
λ(1), and (see [7])
Z ∞
up−1
(2.2)
B(p, q) =
du = B(q, p)
(p, q > 0),
(1 + u)p+q
0
we have (2.1). Suppose for k ∈ N, that (2.1)
Then for k + 1, since
.is valid.
Pk+1
Pk+1 r1 + i=2 ri = λ(k + 1), by setting v = u1
1 + j=2 uj , we obtain
Z
∞
Z
∞
···
(2.3)
0
0
Z
∞
Z
···
=
0
0
1+
∞
1
Pk+1
j=1
k+1
Y
uj
λ(k+1)
r −1
uj j
du1 . . . duk+1
j=1
r 1 Q
P
k+1 rj −1
v r1 −1 1 + k+1
u
j=2 j
j=2 uj
dvdu2 . . . duk+1
λ(k+1)
P
λ(k+1)
1 + k+1
u
(1
+
v)
j=2 j
On a New Multiple Extension of
Hilbert’s Integral Inequality
Bicheng Yang
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Z
∞
Z
∞
···
=
0
0
Qk+1
r −1
uj j
du . . . duk+1
Pk+1 λ(k+1)−r1 2
1 + j=2 uj
Z ∞
v r1 −1
×
dv.
(1 + v)λ(k+1)
0
j=2
In view of (2.2) and the assumption of k, we have
!
Z ∞
k+1
X
v r1 −1
1
(2.4)
dv =
Γ
ri Γ(r1 );
(1 + v)λ(k+1)
Γ(λ(k + 1))
0
i=2
Z
∞
Z
···
(2.5)
0
0
∞
Qk+2
rj −1
Γ(ri )
j=2 uj
du . . . duk+1 = i=2
Pk+1 .
Pk+1 λ(k+1)−r1 2
Γ
1 + j=2 uj
i=2 ri
Qk+1
Then, by (2.5), (2.4) and (2.3), we find
Qk+1 rj −1
Qk+2
Z ∞
Z ∞
j=1 uj
i=1 Γ(ri )
···
du
.
.
.
du
=
.
1
k+1
λ(k+1)
Pk+1
Γ(λ(k + 1))
0
0
1 + j=1 uj
Hence (2.1) is valid for k ∈ N by induction. The lemma is proved.
P
Lemma 2.2. If n ∈ N\{1}, pi > 1 (i = 1, 2, . . . , n), ni=1 p1i = 1 and λ > 0,
set the weight coefficient ω(xi ) as
(λ−pj )/pj
Z ∞ Qn
λZ ∞
pj
j=1(j6=i) xj
(2.6) ω(xi ) := xi
···
dx1 . . . dxi−1 dxi+1 . . . dxn .
P
λ
n
0
0
x
j=1 j
On a New Multiple Extension of
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Bicheng Yang
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Then, each ω(xi ) is constant, that is
n
1 Y
λ
(2.7)
ω(xi ) =
Γ
,
Γ(λ) j=1
pj
(i = 1, 2, . . . , n).
Proof. Fix i. Setting pen = pi , and pej = pj , uj = xj /xi , for j = 1, 2, . . . , i − 1;
pej = pj+1 , uj = xj+1 /xi , for j = i, i + 1, . . . , n − 1 in (2.6), by simplification,
we have
Z ∞
Z ∞
n−1
Y −1+ pλ
1
j
(2.8)
ω(xi ) =
···
uj
du1 . . . dun−1 .
λ
P
n−1
0
0
j=1
1 + j=1 uj
Substitution of n − 1 for k, λ for λ(k) and λ/pej for rj (j = 1, 2, . . . , n) into
(2.1), in view of (2.8), we have
n
n
1 Y
λ
1 Y
λ
ω(xi ) =
Γ
=
Γ
.
Γ(λ) j=1
pej
Γ(λ) j=1
pj
Hence, (2.7) is valid. The lemma is proved.
Lemma 2.3. As in the assumption of Lemma 2.2, for 0 < ε < λ, we have
Z ∞
Z ∞ Qn (λ−pi −ε)/pi
i=1 xi
I := ε
···
P
λ dx1 . . . dxn
n
1
1
j=1 xj
n
1 Y
λ
(2.9)
≥
Γ
(ε → 0+ ).
Γ(λ) i=1
pi
On a New Multiple Extension of
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Proof. Setting ui = xi /xn (i = 1, 2, . . . , n − 1) in the following, we find


Z ∞ Qn−1 (λ−pi −ε)/pi
Z ∞
Z ∞


i=1 ui
···
I=ε
x−1−ε

λ du1 . . . dun−1  dxn
n
P
−1
−1
xn
xn
1
1 + n−1
j=1 uj


Z ∞
Z ∞
Z ∞ Qn−1 (λ−pi −ε)/pi


i=1 ui
≥ε
x−1−ε
···

λ du1 . . . dun−1  dxn
n
P
1
0
0
1 + n−1
j=1 uj
Z ∞
n−1
X
−1
(2.10)
−ε
xn
Aj (xn )dxn ,
1
On a New Multiple Extension of
Hilbert’s Integral Inequality
Bicheng Yang
j=1
where, for j = 1, 2, . . . , n − 1, Aj (xn ) is defined by
Z
Z Qn−1 (λ−pi −ε)/pi
i=1 ui
(2.11)
Aj (xn ) := · · ·
du1 . . . dun−1 ,
Pn−1
λ
u
)
Dj (1 +
j
j=1
satisfying Dj = {(u1 , u2 , . . . , un−1 )|0 < uj ≤ x−1
n , 0 < uk < ∞ (k 6= j)}.
Without loss of generality, we estimate the integral Aj (xn ) for j = 1.
(a) For n = 2, we have
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x−1
n
Z
A1 (xn ) =
0
Z
≤
x−1
n
1
(λ−p −ε)/p1
u1 1
du1
λ
(1 + u1 )
(λ−p1 −ε)/p1
u1
0
du1 =
p1 −(λ−ε)/p1
x
;
λ−ε n
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(b) For n ∈ N\{1, 2}, by (2.1), we have
∞
Z
A1 (xn ) ≤
Z
···
0
0
− λ−ε
p 1 x n p1
≤
λ−ε
=
∞
Z
Qn−1
∞
i=2
···
0
− λ−ε
p 1 x n p1
λ−ε
i
i=2 ui
Pn−1 λ du1 . . . dun−1
1 + j=2 uj
∞
Z
−1+ λ−ε
p
Qn−1
0
Qn
·
i=2
1+
j=2
Γ( λ−ε
)
pi
Γ((λ − ε)(1 − p−1
1 ))
x−1
n
λ−p1 −ε
p1
u1
du1
0
−1+(λ−ε)/pi
ui
Pn−1
Z
uj
du1 . . . dun−1
(λ−ε)(1−p−1
1 )
On a New Multiple Extension of
Hilbert’s Integral Inequality
.
Bicheng Yang
By virtue of the results of (a) and (b), for j = 1, 2, . . . , n − 1,we have
Aj (xn ) ≤
(2.12)
pj −(λ−ε)/pj
xn
Oj (1) (ε → 0+ , n ∈ N\{1}).
λ−ε
+
By (2.11), since for ε → 0 ,
Z
∞
Z
···
0
0
∞
−1+(λ−ε)/pi
i=1 ui
Pn−1 λ du1
1 + j=1 uj
Qn
Qn−1
. . . dun−1 =
i=1 Γ
Γ(λ)
λ
pi
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Z
1
∞
x−1
n
n−1
X
Aj (xn )dxn =
j=1
n−1 Z
X
j=1
∞
x−1
n Aj (xn )dxn
1
Z ∞
n−1
X
pj
j
≤
Oj (1)
x−1−(λ−ε)/p
dxn
n
λ
−
ε
1
j=1
n−1
X p j 2
=
Oj (1),
λ−ε
j=1
then by (2.10) , we find
 Qn
I≥
→
i=1
On a New Multiple Extension of
Hilbert’s Integral Inequality
Bicheng Yang
Γ
λ
pi
Γ(λ)
Qn
λ
i=1 Γ pi
Γ(λ)

+ o(1) − ε
n−1 X
j=1
pj
λ−ε
2
Oj (1)
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(ε → 0+ ).
Thereby, (2.9) is valid and the lemma is proved.
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3.
Proof of the Theorem and Remarks
Proof of Theorem 1.1. By Hölder’s inequality, we have
Z
∞
0
Z
∞
···
=



Z ∞
n
Y
0
≤
0

Z
n 

Y

i=1 




 i=1
∞
∞
Z
···
0
∞
···
J :=
(3.1)
Z
0
P
n
j=1
0
n
Y
1
xj
λ
fi (xi )dx1 . . . dxn
i=1
 p1 
i


λ−pj
n
 (pi −λ)(1−p−1 ) Y pj  
i
x
xj 
 i
  dx1 . . . dxn

j=1


(j6=i)

fi (xi )
P
λ/pi
n
x
j=1 j
fipi (xi )
P
n
j=1
xj
(pi −λ)(1−p−1
i )
λ x i
n
Y
λ−pj
pj
xj
dx1 . . . dxn
1
pi



j=1
.
If (3.1) takes the form of equality, then there exists constants C1 , C2 , . . . , Cn ,
such that they are not all zero and for any i 6= k ∈ {1, 2, . . . , n} (see [5]),
(pi −λ)(1−p−1
i )
fipi (xi )xi
Ci
P
n
j=1 xj
(3.2)
n
Y
λ
j=1
(j6=i)
λ−pj
pj
xj
(p −λ)(1−p−1
k )
fkpk (xk )xk k
= Ck
P
n
j=1 xj
n
Y
λ
λ−pj
pj
xj
Bicheng Yang
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


(j6=i)
On a New Multiple Extension of
Hilbert’s Integral Inequality
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(j6=k)
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a.e. in (0, ∞) × · · · × (0, ∞).
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Assume that Ci 6= 0. By simplification of (3.2), we find
xpi i −λ fipi (xi ) = F (x1 , . . . , xi−1 , xi+1 , . . . , xn )
= constant a.e. in (0, ∞) × · · · × (0, ∞),
R∞
which contradicts the fact that 0 < 0 xpi i −λ−1 fipi (x)dx < ∞. Hence by (2.6)
and (3.1), we conclude
p1
n Z ∞
Y
i
pi −1−λ pi
(3.3)
J<
ω(xi )xi
fi (xi )dxi
.
0
i=1
Then by (2.7), we have (1.5).
For 0 < ε < λ, setting fei (xi ) as: fei (xi ) = 0, for xi ∈ (0, 1);
(λ−p −ε)/pi
fei (xi ) = xi i
,
Bicheng Yang
for xi ∈ [1, ∞) (i = 1, 2, . . . , n),
ε
(3.4)
n Z
Y
∞
xipi −1−λ feipi (xi )dxi
JJ
J
p1
i
= 1.
0
i=1
∞
Z
···
0
II
I
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By (2.9), we find
(3.5) ε
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then we find
Z
On a New Multiple Extension of
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∞
n
Y
1
P
n
j=1
xj
λ
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i=1
Page 14 of 18
n
Y
1
λ
=I≥
Γ
Γ(λ) i=1
pi
(ε → 0+ ).
J. Ineq. Pure and Appl. Math. 6(2) Art. 39, 2005
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If the constant factor
1
Γ(λ)
Qn
i=1 Γ
λ
pi
in (1.5) is not the best possible, then
Qn
1
there exists a positive constant K < Γ(λ) i=1 Γ pλi , such that (1.5) is still
Qn
1
λ
valid if one replaces Γ(λ)
Γ
by K. In particular, one has
i=1
pi
Z
∞
∞
Z
···
ε
0
P
n
j=1
0
n
Y
1
xj
λ
fei (xi )dx1 . . . dxn
i=1
< εK
n Z
Y
i=1
∞
xpi −1−λ feipi (x)dx
On a New Multiple Extension of
Hilbert’s Integral Inequality
p1
i
,
0
Qn
1
λ
and in view of (3.4) and (3.5), it follows that Γ(λ)
Γ
≤ K (ε →
i=1
pi
Qn
λ
1
0+ ). This contradicts the fact K < Γ(λ)
i=1 Γ pi . Hence the constant factor
Qn
1
λ
i=1 Γ pi in (1.5) is the best possible.
Γ(λ)
The theorem is proved.
Remark 1. For λ = 1, inequality (1.7) reduces to (see [10])
Z
∞
Z
(3.6)
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
Bicheng Yang
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For p = q = 2, both (3.6) and (1.2) reduce to (1.1). It follows that inequalities
(3.6) and (1.2) are different extensions of (1.1). Hence, inequality (1.5) is a
new multiple extension of (1.1). Since all the constant factors in the obtained
inequalities are the best possible, we have obtained new results.
On a New Multiple Extension of
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, 1952.
[3] YONG HONG, All-sided generalization about Hardy-Hilbert’s integral inequalities, Acta Math. Sinica, 44(4) (2001), 619–626.
[4] LEPING HE, JIANGMING YU AND MINGZHE GAO, An extension of
Hilbert’s integral inequality, J. Shaoguan University (Natural Science),
23(3) (YEAR), 25–29.
[5] JICHANG KUANG, Applied Inequalities, Hunan Education Press,
Changsha, 2004.
[6] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Boston, 1991.
[7] ZHUXI WANG AND DUNRIN GUO, An Introduction to Special Functions, Science Press, Beijing, 1979.
[8] BICHENG YANG, On a multiple Hardy-Hilbert’s integral inequality, Chinese Annals of Math., 24A(6) (2003), 743–750.
[9] BICHENG YANG, On Hardy-Hilbert’s integral inequality, J. Math. Anal.
Appl., 261 (2001), 295–306.
On a New Multiple Extension of
Hilbert’s Integral Inequality
Bicheng Yang
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[10] BICHENG YANG, On the extended Hilbert’s integral inequality, J. Inequal. Pure and Appl. Math., 5(4) (2004), Art. 96. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=451]
[11] BICHENG YANG AND Th. M. RASSIAS, On the way of weight coefficient and research for the Hilbert-type inequalities, Math. Ineq. Appl., 6(4)
(2003), 625–658.
On a New Multiple Extension of
Hilbert’s Integral Inequality
Bicheng Yang
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