Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 4, Article 153, 2006
BEST GENERALIZATION OF HARDY-HILBERT’S INEQUALITY WITH
MULTI-PARAMETERS
DONGMEI XIN
D EPARTMENT OF M ATHEMATICS
G UANGDONG E DUCATION C OLLEGE
G UANGZHOU , G UANGDONG 510303
P EOPLE ’ S R EPUBLIC O F C HINA .
xdm77108@gdei.edu.cn
Received 07 February, 2006; accepted 13 April, 2006
Communicated by B. Yang
A BSTRACT. 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.
Key words and phrases: Hardy-Hilbert’s inequality, Hölder’s inequality, weight function, β function.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
If p > 1, p1 + 1q = 1, f, g are non-negative functions such that 0 <
R∞
0 < 0 g q (t)dt < ∞, then we have
Z
∞
Z
(1.1)
0
0
∞
f (x)g(y)
π
dxdy <
x+y
sin( πp )
Z
∞
Z
(1.2)
0
0
∞
f (x)
dx
x+y
p
Z
∞
p1 Z
f p (t)dt
π
dy <
sin( πp )
0
f p (t)dt < ∞ and
1q
g q (t)dt ;
0
0
"
∞
R∞
#p Z
∞
f p (t)dt,
0
h
ip
π
π
where the constant factors sin(π/p)
and sin(π/p)
are the best possible (see [1]). 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).
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
034-06
2
D ONGMEI X IN
In 2002, Yang [3] gave some generalizations of (1.1) and (1.2) by introducing a parameter
λ > 0 as:
∞
Z
(1.3)
0
Z
∞
f (x)g(y)
dxdy
xλ + y λ
0
Z ∞
p1 Z ∞
1q
π
(q−1)(1−λ) q
(p−1)(1−λ) p
<
t
f (t)dt
t
g (t)dt ;
λ sin( πp )
0
0
∞
Z
y λ−1
(1.4)
∞
Z
0
0
f (x)
dx
λ
x + yλ
p
"
π
dy <
λ sin( πp )
#p Z
∞
t(p−1)(1−λ) f p (t)dt,
0
h
ip
π
π
where the constant factors λ sin(π/p)
and λ sin(π/p)
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
(1.5)
0
Z
∞
y
(1.6)
Z
∞
f (x)g(y)
dxdy
xλ + y λ
0
Z ∞
p1 Z ∞
1q
λ
π
p(1− λ
)−1
p
q(1−
)−1
q
r
s
<
f (x)dx
x
x
g (x)dx ;
λ sin( πr )
0
0
pλ
−1
s
Z
0
∞
0
f (x)
dx
λ
x + yλ
p
π
dy <
λ sin( πr )
p Z
∞
λ
xp(1− r )−1 f p (x)dx,
0
h
ip
π
π
and λ sin(π/r)
are the best possible. Inequality (1.5) is
where the constant factors λ sin(π/r)
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]):
#2 Z
"
1q
p1 Z ∞
Z ∞ Z ∞ ln x f (x)g(y)
∞
y
π
g q (t)dt ;
(1.7)
dxdy <
f p (t)dt
x−y
sin( πp )
0
0
0
0
Z
(1.8)
∞


0
ln
x
y
f (x)
x−y
p
"
π
dx dy <
sin( πp )
#2p Z
∞
f p (t)dt,
0
i2
h
i2p
h
π
π
where the constant factors sin(π/p)
and sin(π/p)
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).
J. Inequal. Pure and Appl. Math., 7(4) Art. 153, 2006
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H ARDY-H ILBERT ’ S I NEQUALITY W ITH M ULTI -PARAMETERS
3
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 ∞ ln x f (x)g(y)
y
(1.9)
dxdy
xλ − y λ
0
0
"
#2 Z
p1 Z ∞
1q
∞
π
(q−1)(1−λ) q
(p−1)(1−λ) p
t
g (t)dt ;
<
t
f (t)dt
λ sin( πp )
0
0
Z
(1.10)
0
∞

y λ−1 
ln
x
y
f (x)
xλ − y λ
p
"
π
dx dy <
λ sin( πp )
#2p Z
∞
t(p−1)(1−λ) f p (t)dt,
0
h
i2
h
i2p
π
π
where the constant factors λ sin(π/p)
and λ sin(π/p)
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.
2. S OME L EMMAS
Lemma 2.1. If p > 1, p1 + 1q = 1, r > 1, 1s + 1r = 1, λ > 0, define the weight function ωλ (s, p, x)
as
Z ∞ ln x
λ
y
x(p−1)(1− r )
(2.1)
ωλ (s, p, x) =
·
dy, x ∈ (0, ∞).
λ
xλ − y λ
y 1− s
0
Then we have
ωλ (s, p, x) = x
(2.2)
)−1
p(1− λ
r
π
λ sin( πr )
2
.
Proof. For fixed x, setting u = ( xy )λ in the integral (2.1) and by [1] (see [1, Th. 342 Remark]),
we have
Z ∞
λ
1
ln u
x(p−1)(1− r ) 1 −1
(2.3)
xu λ du
ωλ (s, p, x) = 2
·
λ 0 xλ (u − 1) (u λ1 x)1− λs
Z
1
1 p(1− λ )−1 ∞ ln u
r
· u− r du
= 2x
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.
Note. By (2.3), we still have
Z
(2.4)
ωλ (r, q, y) =
0
∞
ln
xλ − y λ
λ
= y q(1− s )−1
J. Inequal. Pure and Appl. Math., 7(4) Art. 153, 2006
x
y
λ
·
x(q−1)(1− s )
λ
y 1− r
2
π
.
λ sin( πs )
dx
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D ONGMEI X IN
3. M AIN R ESULTS AND A PPLICATIONS
Theorem 3.1. If p > 1, p1 + 1q = 1, r > 1, 1s + 1r = 1, λ > 0, f, g ≥ 0 such that 0 <
R ∞ p(1− λ )−1 p
R∞
λ
r
x
f (x)dx < ∞, and 0 < 0 xq(1− s )−1 g q (x)dx < ∞, then we have
0
Z
∞
∞
Z
ln
x
y
f (x)g(y)
dxdy
xλ − y λ
p1 Z ∞
1q
2 Z ∞
π
)−1 p
q(1− λs )−1 q
p(1− λ
r
<
f (x)dx
x
g (x)dx ,
x
λ sin( πr )
0
0
(3.1)
0
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
∞
(3.2)
0
0
<
x
y
f (x)g(y)
xλ − y λ
π 2 Z ∞
λ
dxdy
x
p1 Z
p(1− λ
)−1 p
2
∞
f (x)dx
x
0
)−1 q
q(1− λ
2
g (x)dx
1q
,
0
(b) for λ = 1, we have
∞
Z
Z
∞
ln
x
y
0
f (x)g(y)
dxdy
x−y
2 Z ∞
p1 Z ∞
1q
p
q
π
−1
p
−1
q
<
x s f (x)dx
x r g (x)dx .
sin( πr )
0
0
(3.3)
0
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

  1
p
q



x
x
Z ∞Z ∞
λ
λ
  ln y

 ln y
(1− r )/q
(1− s )/p
x
y



 ·
f
(x)
g(y)
dxdy
=
·
λ
λ
λ
λ
λ
λ



y (1− s )/p
x(1− r )/q
0
0

 x −y

 x −y
≤

Z
∞

Z

0


Z

=
xλ
0
×
Z
ln
∞
x
y
−
yλ

∞ Z

0
0
·
∞
x
(p−1)(1− λ
)
r
(1− λs )
ln
xλ
∞
p
y
x
y
−
yλ
·
p1 Z
J. Inequal. Pure and Appl. Math., 7(4) Art. 153, 2006
dy  f p (x)dx
 p1


y
(q−1)(1− λs )
λ
x(1− r )

dx g q (y)dy
 1q


∞
q
ωλ (r, q, y)g (y)dy
ωλ (s, p, x)f (x)dx
0

1q
.
0
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H ARDY-H ILBERT ’ S I NEQUALITY W ITH M ULTI -PARAMETERS
5
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
)
(p−1)(1− λ
ln xy
ln
r
y
x
y (q−1)(1− s ) q
p
A λ
f (x) = B λ
g (y),
·
·
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),
a.e. in (0, ∞).
)−1
p(1− λ
r
Without loss of generality, suppose A 6= 0, we may get x
f p (x) = C/(Ax), a.e. in
R ∞ p(1− λ )−1 p
r
(0, ∞), which contradicts 0 < 0 x
f (x)dx < ∞. Hence (3.4) takes strict inequality
as follows:
Z ∞ Z ∞ ln x f (x)g(y)
y
(3.5)
dxdy
xλ − y λ
0
0
Z ∞
p1 Z ∞
1q
p
q
<
ωλ (s, p, x)f (x)dx
ωλ (r, q, y)g (y)dy
.
0
0
In view of (2.2) and (2.4),
h we have
i (3.1).
2
π
in (3.1) is not the best possible, then there exists a positive
If the constant factor λ sin(π/r)
h
i2 π
constant K with K < λ sin(π/r)
and an a > 0. We have
Z
∞
∞
Z
ln
x
y
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
For ε > 0 small enough ε <
pλ
r
then we find
Z ∞Z
(3.7)
a
∞
ln
x
y
g (x)dx
.
and 0 < b < a, setting fε and gε as:
−1− pε + λ
r
x ∈ (0, b);
,
ε
λ
gε = x−1− q + s ,
fε (x) · gε (y)
xλ − y λ
b
x
1q
q(1− λs )−1 q
a
fε (x) = gε (x) = 0,
fε = x
∞
∞
Z
∞
Z
ln
x ∈ [b, ∞),
dxdy =
a
x
y
ε
λ
ε
λ
x−1− p + r · y −1− q + s
dxdy.
xλ − y λ
b
In (3.7), for b → 0+ , by (3.6), we have
1
λ 2 aε
Z
∞
ln u −1+ 1s − qλε
du = ε
u
u−1
Z
∞
Z
∞
ln
x
y
fε (x)gε (y)
dxdy ≤
K
.
aε
xλ − y λ
i2
h
π
+
For ε → 0, by [1] (see [1, Th. 342 Remark]), it follows that λ sin(π/r) ≤ K, which contrah
i2
h
i2
π
π
dicts the fact that K < λ sin(π/r)
. Hence the constant factor λ sin(π/r)
in (3.1) is the best
possible. The theorem is proved.
0
J. Inequal. Pure and Appl. Math., 7(4) Art. 153, 2006
a
0
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6
D ONGMEI X IN
Theorem 3.2. If p > 1, p1 + 1q = 1, r > 1,
R ∞ p(1− λ )−1 p
r
x
f (x)dx < ∞, then we have
0

Z
∞
pλ
−1 
s
y
Z
(3.8)
0
ln
∞
h
x
y
xλ
0
where the constant
particular,
π
λ sin(π/r)
i2p
−
1
s
p
f (x)
dx dy <
yλ
1
r
+
= 1, λ > 0, f ≥ 0 such that 0 <
π
λ sin( πr )
2p Z
∞
λ
xp(1− r )−1 f p (x)dx,
0
is the best possible. Inequality (3.8) is equivalent to (3.1). In
(a) for r = s = 2, we have
Z
(3.9)

Z
∞
pλ
−1 
2
y
∞
ln
x
y
f (x)
xλ − y λ
0
0
p
π 2p Z ∞
λ

dx dy <
xp(1− 2 )−1 f p (x)dx,
λ
0
(b) for λ = 1, we have
Z
∞
(3.10)

Z
p
−1 
s
y
0
0
∞
ln
x
y
p
f (x)
x−y
dx dy <
π
sin( πr )
2p Z
∞
p
x s −1 f p (x)dx.
0
Proof. Setting a real function g(y) as

Z
pλ
−1

g(y) = y s
0
∞
ln
x
y
p−1
f (x)
xλ − y λ
dx
,
y ∈ (0, ∞),
then by (3.1), we find
Z ∞
p
q(1− λs )−1 q
(3.11)
y
g (y)dy
0


p p
Z ∞ ln x f (x)
Z ∞ pλ

y
 dy
=
y s −1 
dx
 0

xλ − y λ
0

p
Z ∞ Z ∞ ln x f (x)g(y)
y
=
dxdy 
xλ − y λ
0
0
Z ∞
p−1
2p Z ∞
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
Hence we obtain
Z
(3.12)
0<
∞
y
0
π
g (y)dy ≤
λ sin( πr )
q(1− λs )−1 q
2p Z
∞
λ
xp(1− r )−1 f p (x)dx < ∞.
0
By (3.1), both (3.11) and (3.12) take the form of strict inequality, and we have (3.8).
J. Inequal. Pure and Appl. Math., 7(4) Art. 153, 2006
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H ARDY-H ILBERT ’ S I NEQUALITY W ITH M ULTI -PARAMETERS
7
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 ∞ ln x f (x)
Z ∞
h λ 1
i
1
λ
y
 y − s + p g(y) dy
y s − p
dx
=
xλ − y λ
0
0
≤

Z

0
∞

Z
pλ
−1 
s
y
∞
ln
0
x
y
f (x)
xλ − y λ
p
dx dy
 p1
 Z

∞
y
q(1− λs )−1 q
g (y)dy
1q
.
0
Then by (3.8), we have (3.1). Hence (3.1) and (3.8) are equivalent.
h
i2p
π
If the constant λ sin(π/r)
in (3.8) is not the best possible, by using (3.13), 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 3.3.
(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.
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