Journal of Inequalities in Pure and
Applied Mathematics
BEST GENERALIZATION OF HARDY-HILBERT’S INEQUALITY
WITH MULTI-PARAMETERS
volume 7, issue 4, article 153,
2006.
DONGMEI XIN
Department of Mathematics
Guangdong Education College
Guangzhou, Guangdong 510303
People’s Republic Of China.
Received 07 February, 2006;
accepted 13 April, 2006.
Communicated by: B. Yang
EMail: xdm77108@gdei.edu.cn
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
034-06
Quit
Abstract
By introducing some parameters and the β function and improving the weight
function, we obtain a generalization of Hilbert’s integral inequality with the best
constant factor. As its applications, we build its equivalent form and some particular results.
2000 Mathematics Subject Classification: 26D15.
Key words: Hardy-Hilbert’s inequality, Hölder’s inequality, weight function, β function.
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Results and Applications . . . . . . . . . . . . . . . . . . . . . . . . .
References
Dongmei Xin
3
7
9
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
1.
Introduction
If p > 1, p1 + 1q = 1, f, g are non-negative functions such that 0 <
R∞
∞ and 0 < 0 g q (t)dt < ∞, then we have
Z
∞
∞
Z
(1.1)
0
0
π
f (x)g(y)
dxdy <
x+y
sin( πp )
Z
∞
p1 Z
f (t)dt
p
0
R∞
∞
0
f p (t)dt <
1q
g (t)dt ;
q
0
#p Z
∞
π
f p (t)dt,
dy <
(1.2)
π
sin( p )
0
0
0
ip
h
π
π
are the best possible (see [1]).
where the constant factors sin(π/p)
and sin(π/p)
Inequality (1.1) is well known as Hardy-Hilbert’s integral inequality, which is
important in analysis and applications (see [2]). Inequality (1.1) is equivalent to
(1.2).
In 2002, Yang [3] gave some generalizations of (1.1) and (1.2) by introducing
a parameter λ > 0 as:
Z ∞Z ∞
f (x)g(y)
(1.3)
dxdy
xλ + y λ
0
0
Z ∞
p1
π
(p−1)(1−λ) p
<
t
f (t)dt
λ sin( πp )
0
Z ∞
1q
(q−1)(1−λ) q
×
t
g (t)dt ;
Z
∞
Z
∞
f (x)
dx
x+y
p
"
0
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
Z
∞
y
(1.4)
λ−1
Z
0
∞
0
f (x)
dx
λ
x + yλ
p
dy
"
π
<
λ sin( πp )
h
π
λ sin(π/p)
#p Z
π
λ sin(π/p)
∞
t(p−1)(1−λ) f p (t)dt,
0
ip
where the constant factors
and
are the best possible. Inequality (1.3) is equivalent to (1.4).
When λ = 1, both (1.3) and (1.4) change to (1.1) and (1.2). Yang [4] gave
another generalization of (1.1) by introducing a parameter λ and a β function.
In 2004, by introducing some parameters and estimating the weight function,
Yang [5] gave some extensions of (1.1) and (1.2) with the best constant factors
as:
Z ∞Z ∞
f (x)g(y)
(1.5)
dxdy
xλ + y λ
0
0
Z ∞
p1 Z ∞
1q
λ
π
p(1− λ
)−1
p
q(1−
)−1
q
r
s
<
x
f (x)dx
x
g (x)dx ;
λ sin( πr )
0
0
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Z
∞
y
(1.6)
0
pλ
−1
s
Z
0
∞
p
f (x)
dx dy
+ yλ
p Z ∞
λ
π
<
xp(1− r )−1 f p (x)dx,
π
λ sin( r )
0
xλ
Close
Quit
Page 4 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
h
ip
π
π
where the constant factors λ sin(π/r)
and λ sin(π/r)
are the best possible. Inequality (1.5) is equivalent to (1.6). Recently, [6, 7, 8, 9] considered some
multiple extensions of (1.1).
Under the same conditions with (1.1), we still have (see [1, Th. 342]):
Z
∞Z
∞
ln
x
y
(1.7)
0
f (x)g(y)
dxdy
#2 Z
"
p1 Z ∞
1q
∞
π
q
p
f (t)dt
g (t)dt ;
<
sin( πp )
0
0
x−y
0
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
∞
Z
(1.8)

ln

0
x
y
f (x)
x−y
h
p
"
π
dx dy <
sin( πp )
π
sin(π/p)
i2
h
π
sin(π/p)
#2p Z
i2p
∞
f p (t)dt,
0
where the constant factors
and
are the best possible. Inequality (1.7) is equivalent to (1.8). In recent years, by introducing a parameter
λ, Kuang [10] gave an new extension of (1.7).
In 2003, by introducing a parameter λ > 0 and the weight function, Yang
[11] gave another generalisation of (1.7) and the extended equivalent form as:
Z
∞
Z
(1.9)
0
0
∞
ln
x
y
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 18
f (x)g(y)
xλ − y λ
Title Page
dxdy
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
"
π
<
λ sin( πp )
Z
(1.10)
0
∞
# 2 Z
p1 Z
f (t)dt
∞
t
0

y λ−1 
ln
∞
(p−1)(1−λ) p
1q
g (t)dt ;
(q−1)(1−λ) q
t
0
x
y
p
f (x)
xλ − y λ
dx dy
"
π
<
λ sin( πp )
h
π
λ sin(π/p)
i2
h
#2p Z
∞
t(p−1)(1−λ) f p (t)dt,
0
π
λ sin(π/p)
i2p
where the constant factors
and
are the best possible.
Inequality (1.9) is equivalent to (1.10).
In this paper, by using the β function and obtaining the expression of the
weight function, we give a new extension of (1.7) with some parameters as
(1.5). As applications, we also consider the equivalent form and some other
particular results.
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
2.
Some Lemmas
Lemma 2.1. If p > 1,
function ωλ (s, p, x) as
1
p
+
Z
(2.1)
ωλ (s, p, x) =
0
1
q
∞
= 1, r > 1,
ln
1
s
+
x
y
xλ − y λ
1
r
= 1, λ > 0, define the weight
λ
·
x(p−1)(1− r )
λ
y 1− s
dy,
x ∈ (0, ∞).
Then we have
(2.2)
ωλ (s, p, x) = x
)−1
p(1− λ
r
π
λ sin( πr )
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
2
.
Proof. For fixed x, setting u = ( xy )λ in the integral (2.1) and by [1] (see [1, Th.
342 Remark]), we have
Dongmei Xin
Title Page
Contents
(2.3)
Z ∞
λ
1
ln u
x(p−1)(1− r ) 1 −1
ωλ (s, p, x) = 2
·
xu λ du
λ 0 xλ (u − 1) (u λ1 x)1− λs
Z
1
1 p(1− λ )−1 ∞ ln u
r
= 2x
· u− r du
λ
u−1
0
2
λ
π
=
xp(1− r )−1 .
π
λ sin( r )
Hence, (2.2) is valid and the lemma is proved.
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
Note. By (2.3), we still have
Z
(2.4)
ωλ (r, q, y) =
0
∞
ln
x
y
xλ − y λ
λ
= y q(1− s )−1
λ
·
x(q−1)(1− s )
λ
y 1− r
2
π
.
λ sin( πs )
dx
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
3.
Main Results and Applications
Theorem 3.1. If p > 1, p1 + 1q = 1, r > 1, 1s + 1r = 1, λ > 0, f, g ≥ 0 such that
R∞
R∞
λ
λ
0 < 0 xp(1− r )−1 f p (x)dx < ∞, and 0 < 0 xq(1− s )−1 g q (x)dx < ∞, then we
have
Z ∞ Z ∞ ln x f (x)g(y)
y
(3.1)
dxdy
xλ − y λ
0
0
p1
2 Z ∞
π
)−1
p
p(1− λ
r
f (x)dx
<
x
λ sin( πr )
0
Z ∞
1q
q(1− λs )−1 q
×
x
g (x)dx ,
0
where the constant factor
h
π
λ sin(π/r)
i2
is the best possible. In particular,
(a) for r = s = 2, we have
Z ∞ Z ∞ ln x f (x)g(y)
y
(3.2)
dxdy
xλ − y λ
0
0
p1
π 2 Z ∞
p(1− λ
)−1
p
2
<
x
f (x)dx
λ
0
Z ∞
1q
q(1− λ
)−1
q
2
×
x
g (x)dx ,
0
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
(b) for λ = 1, we have
Z
∞
Z
∞
ln
(3.3)
0
0
π
<
sin( πr )
x
y
f (x)g(y)
dxdy
x−y
2 Z ∞
x
p
−1
s
p1 Z
p
∞
f (x)dx
x
0
q
−1
r
q
1q
g (x)dx
Proof. By Hölder’s inequality and Lemma 2.1, we have
Z ∞ Z ∞ ln x f (x)g(y)
y
(3.4)
dxdy
xλ − y λ
0
0

  1
p

x
Z ∞Z ∞
 ln y

)/q
(1− λ
r
x

 ·
f
(x)
=
λ
λ
λ


y (1− s )/p
0
0

 x −y
  1

q


 ln xy

(1− λs )/p
y


×
·
g(y) dxdy
λ
λ
λ


x(1− r )/q
 x −y

≤

Z

∞

Z

0
0
×
ln
xλ
yλ
∞

Z

0
x
y
−

∞ Z

0
.
0
·
∞
x
(p−1)(1− λ
)
r
(1− λs )
ln
y
x
y
xλ − y λ

dy  f p (x)dx
Dongmei Xin
Title Page
Contents
JJ
J
y (q−1)(1− s )
λ
x(1− r )

dx g q (y)dy
II
I
Go Back
 p1

Close

λ
·
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Quit
 1q


Page 10 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
∞
Z
p
=
p1 Z
ωλ (s, p, x)f (x)dx
0
1q
∞
q
ωλ (r, q, y)g (y)dy
.
0
If (3.4) takes the form of equality, then there exist constants A and B, such that
they are not all zero and (see [12])
x
λ
λ
ln xy
ln y
y (q−1)(1− s ) q
x(p−1)(1− r ) p
f (x) = B λ
·
g (y),
A λ
·
λ
λ
x − yλ
x − yλ
y 1− s
x1− r
a.e. in (0, ∞) × (0, ∞).
λ
λ
We find that Ax·xp(1− r )−1 f p (x) = By ·y q(1− s )−1 g q (y), a.e. in (0, ∞)×(0, ∞).
Hence there exists a constant C, such that
λ
λ
Ax · xp(1− r )−1 f p (x) = C = By · y q(1− s )−1 g q (y),
λ
∞
Z
(3.5)
0
0
∞
ln
x
y
dxdy
xλ − y λ
Z ∞
p1 Z
p
<
ωλ (s, p, x)f (x)dx
In view of (2.2) and (2.4), we have (3.1).
Contents
JJ
J
II
I
Go Back
f (x)g(y)
0
Dongmei Xin
Title Page
a.e. in (0, ∞).
Without loss of generality, suppose A 6= 0, we may get xp(1− r )−1 f p (x) =
R∞
λ
C/(Ax), a.e. in (0, ∞), which contradicts 0 < 0 xp(1− r )−1 f p (x)dx < ∞.
Hence (3.4) takes strict inequality as follows:
Z
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Close
∞
q
ωλ (r, q, y)g (y)dy
Quit
1q
.
Page 11 of 18
0
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
If the constant factor
h
π
λ sin(π/r)
exists a positive constant K
∞
Z
∞
Z
ln
x
y
i2
in (3.1) is not the best possible, then there
i2 h
π
with K < λ sin(π/r)
and an a > 0. We have
f (x)g(y)
dxdy
xλ − y λ
Z ∞
p1 Z
p(1− λ
)−1
p
r
<K
x
f (x)dx
(3.6)
a
0
a
x
1q
q(1− λs )−1 q
g (x)dx
.
a
For ε > 0 small enough ε <
pλ
r
and 0 < b < a, setting fε and gε as:
−1− pε + λ
r
Dongmei Xin
ε
λ
gε = x−1− q + s ,
,
x ∈ [b, ∞),
Title Page
Contents
then we find
∞
Z
Z
(3.7)
a
b
∞
ln
x
y
JJ
J
fε (x) · gε (y)
dxdy
xλ − y λ
Z
∞
Z
ln
∞
=
a
x
y
ε
λ
ε
xλ − y λ
b
1
λ 2 aε
0
∞
ln u −1+ 1s − qλε
u
du = ε
u−1
Z
a
II
I
Go Back
λ
x−1− p + r · y −1− q + s
dxdy.
Close
Quit
In (3.7), for b → 0+ , by (3.6), we have
Z
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
x ∈ (0, b);
fε (x) = gε (x) = 0,
fε = x
∞
∞
Z
0
∞
ln
x
y
Page 12 of 18
fε (x)gε (y)
xλ
−
yλ
dxdy ≤
K
.
aε
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
i2
h
π
For ε+ → 0, by [1] (see [1, Th. 342 Remark]), it follows that λ sin(π/r)
≤ K,
h
i2
π
which contradicts the fact that K < λ sin(π/r)
. Hence the constant factor
h
i2
π
in (3.1) is the best possible. The theorem is proved.
λ sin(π/r)
Theorem 3.2. If p > 1, p1 + 1q = 1, r > 1, 1s +
R∞
λ
0 < 0 xp(1− r )−1 f p (x)dx < ∞, then we have
Z
(3.8)

Z
∞
pλ
−1
ys 
∞
h
x
y
ln
π
λ sin(π/r)
where the constant
lent to (3.1). In particular,
f (x)
xλ − y λ
<
0
0
i2p
1
r
= 1, λ > 0, f ≥ 0 such that
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
p
dx dy
π
λ sin( πr )
Dongmei Xin
2p Z
∞
λ
xp(1− r )−1 f p (x)dx,
Contents
is the best possible. Inequality (3.8) is equiva-
(a) for r = s = 2, we have
Z
(3.9)
0
∞
Title Page
0
JJ
J
II
I
Go Back

Z
pλ
−1 
2
y
0
∞
ln
x
y
p
Close
dx dy
Quit
f (x)
xλ − y λ
<
π 2p Z
λ
0
Page 13 of 18
∞
x
p(1− λ
)−1 p
2
f (x)dx,
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
(b) for λ = 1, we have
Z
(3.10)
0
∞

Z
p
−1
ys 
0
∞
ln
x
y
f (x)
p
dx dy
x−y
π
<
sin( πr )
2p Z
Proof. Setting a real function g(y) as

p−1
Z ∞ ln x f (x)
pλ
y
dx ,
g(y) = y s −1 
xλ − y λ
0
∞
p
x s −1 f p (x)dx.
0
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
y ∈ (0, ∞),
then by (3.1), we find
Z ∞
p
q(1− λs )−1 q
(3.11)
g (y)dy
y
0


p p
Z ∞ ln x f (x)
Z ∞ pλ

y
−1 

s
=
dx dy
y
 0

xλ − y λ
0

p
Z ∞ Z ∞ ln x f (x)g(y)
y
=
dxdy 
λ − yλ
x
0
0
2p Z ∞
π
p(1− λ
)−1
p
r
≤
x
f (x)dx
λ sin( πr )
0
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
∞
Z
×
x
p−1
q(1− λs )−1 q
g (x)dx
.
0
Hence we obtain
Z
(3.12)
0<
∞
λ
y q(1− s )−1 g q (y)dy
2p Z ∞
λ
π
xp(1− r )−1 f p (x)dx < ∞.
≤
π
λ sin( r )
0
0
By (3.1), both (3.11) and (3.12) take the form of strict inequality, and we have
(3.8).
On the other hand, suppose that (3.8) is valid. By Hölder’s inequality, we
find
Z ∞ Z ∞ ln x f (x)g(y)
y
(3.13)
dxdy
xλ − y λ
0
0


Z ∞
Z ∞ ln x f (x)
h λ 1
i
λ
1
y
y s − p
 y − s + p g(y) dy
=
dx
xλ − y λ
0
0
≤

Z


Z
∞
pλ
−1
ys 
∞
0
0
Z
×
∞
y
0
ln
x
y
p
f (x)
xλ − y λ
q(1− λs )−1 q
dx dy
g (y)dy
 p1

Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit

Page 15 of 18
1q
.
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
Then by (3.8), we have (3.1). Hence (3.1) and (3.8) are equivalent.
h
i2p
π
in (3.8) is not the best possible, by using (3.13),
If the constant λ sin(π/r)
we may get a contradiction that the constant factor in (3.1) is not the best possible. Thus we complete the proof of the theorem.
Remark 1.
(a) For r = q, s = p, Inequality (3.1) reduces to (1.9) and (3.8) reduces to
(1.10).
(b) Inequality (3.1) is an extension of (1.7) with parameters (λ, r, s).
(c) It is interesting that inequalities (1.9) and (3.2) are different, although they
have the same parameters and possess a best constant factor.
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
References
[1] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge University Press, Cambridge, UK, 1952.
[2] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Boston, 1991.
[3] BICHENG YANG, On an extension of Hardy-Hilbert’s inequality, J.
Math. Anal. Appl., 23A(2) (2002), 247–254.
[4] BICHENG YANG, On Hardy-Hilbert’s integral inequality, J. Math. Anal.
Appl., 261 (2001), 295–306.
[5] BICHENG YANG, On the extended Hilbert’s integral inequality and applications, International Review of Pure and Applied Mathematics, (2005),
10–11.
[6] YONG HONG , All-sided generalization about Hardy-Hilbert’s integral
inequalities, Acta Math. Sinica, 44(4) (2001), 619–626.
[7] LEPING HE, JIANGMING YU AND MINZHE GAO, An extension of
Hilbert’s integral inequality, J. Shaoguan Univ., (Natural Science), 23(3)
(2002), 25–30.
[8] JICHANG KUANG, On new extensions of Hilbert’s integral inequality, J.
Math. Anal. Appl., 235 (1999), 608–614.
[9] M. KRNIĆ AND J. PEČARIĆ, General Hilbert’s and Hardy’s inequalities,
Math. Inequal. & Applics., 8(1) (2005), 29–51.
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au
[10] JICHANG KUANG AND L. DEBNATH, On new generalizations of
Hilbert’s inequality and their applications, J. Math. Anal. Appl., 245
(2000), 248–265.
[11] BICHENG YANG, On a generalization of a Hilbert’s type integral inequality and its applications, Mathematics Applications, 16(2) (2003), 82–86.
[12] JICHANG KUANG, Applied Inequalities, Shandong Science and Technology Press, Jiinan, (2004).
Best Generalization of
Hardy-Hilbert’s Inequality With
Multi-Parameters
Dongmei Xin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 18
J. Ineq. Pure and Appl. Math. 7(4) Art. 153, 2006
http://jipam.vu.edu.au