Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 4, Article 92, 2005
ON HARDY-HILBERT INTEGRAL INEQUALITIES WITH SOME PARAMETERS
YONG HONG
D EPARTMENT OF M ATHEMATICS
G UANGDONG B USINESS C OLLEGE ,
G UANGZHOU 510320
P EOPLE ’ S R EPUBLIC OF C HINA
hongyong59@sohu.com
Received 07 April, 2005; accepted 06 June, 2005
Communicated by B. Yang
A BSTRACT. In this paper, we give a new Hardy-Hilbert’s integral inequality with some parameters and a best constant factor. It includes an overwhelming majority of results of many papers.
Key words and phrases: Hardy-Hilbert’s integral inequality, Weight, Parameter, Best constant factor, β-function, Γ-function.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION AND M AIN R ESULT
R∞
R∞
Let p1 + 1q = 1(p > 1), f ≥ 0, g ≥ 0, 0 < 0 f p (x)dx < +∞, 0 < 0 g q (x)dx < +∞,
then we have the well known Hardy-Hilbert inequality
Z ∞
p1 Z ∞
1q
Z ∞Z ∞
f (x)g(x)
π
p
q
f (x)dx
g (x)dx ;
dxdy <
(1.1)
π
x+y
0
0
0
0
sin p
and an equivalent form as:
Z ∞ Z
(1.2)
0
0
∞
f (x)
dx
x+y
p
p

dy < 
sin
π 
π
p
Z
∞
f p (x)dx.
0
In recent years, many results have been obtained in the research of these two inequalities (see
[1] – [13]). Yang [1] and [2] gave:
Z ∞Z ∞
f (x)g(y)
p+λ−2 p+λ−2
(1.3)
dxdy < B
,
(x + y)λ
p
q
0
0
Z ∞
p1 Z ∞
1q
x1−λ g q (x)dx ,
×
x1−λ f p (x)dx
0
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
109-05
0
2
YONG H ONG
where B(r, s) is the β-function; and Kuang [3] gave:
Z ∞Z ∞
f (x)g(x)
(1.4)
dxdy
xλ + y λ
0
0
Z ∞
p1 Z ∞
1q
π
1−λ q
1−λ p
1 x g (x)dx .
x f (x)dx
<
1
π
π
p
0
0
sin q qλ
λ sin pλ
Recently, Hong [4] gave:
Z ∞Z ∞
f (x)g(x)
p
dxdy
(1.5)
x2 + y 2
0
0
1
≤ √ Γ
2 π
1
2p
Z ∞
p1 Z ∞
1q
1
p
q
Γ
f (x)dx
g (x)dx .
2q
0
0
And Yang [5] gave:
Z ∞Z ∞
f (x)g(x)
(1.6)
dxdy
xλ + y λ
0
0
π
<
λ sin πp
∞
Z
y
(1.7)
0
λ−1
Z
0
∞
p1 Z
∞
Z
x
(p−1)(1−λ) p
f (x)dx
x
0
f (x)
dx
λ
x + yλ
1q
∞
(q−1)(1−λ) q
g (x)dx
;
0
p
p

dy < 
∞
Z
π

λ sin πp
x(p−1)(1−λ) f p (x)dx.
0
These results generalize and improve (1.1) and (1.2) in a certain degree.
In this paper, by introducing a few parameters, we obtain a new Hardy-Hilbert integral inequality with a best constant factor, which is a more extended inequality, and includes all the
results above and the overwhelming majority of results of many recent papers.
Our main result is as follows:
Theorem 1.1. If p1 + 1q = 1 (p > 1), α > 0, λ > 0, m, n ∈ R, such that 0 < 1 − mp < αλ,
0 < 1 − nq < αλ, and f ≥ 0, g ≥ 0, satisfy
Z ∞
(1.8)
0<
x(1−αλ)+p(n−m) f p (x)dx < ∞,
0
Z
0<
(1.9)
∞
y (1−αλ)+q(m−n) g q (y)dy < ∞,
0
then
Z
∞
Z
(1.10)
0
0
∞
f (x)g(x)
dxdy < Hλ,α (m, n, p, q)
(xα + y α )λ
Z
p1
∞
x
(1−αλ)+p(n−m) p
f (x)dx
0
Z
×
∞
y
(1−αλ)+q(m−n) q
g (y)dy
1q
;
0
J. Inequal. Pure and Appl. Math., 6(4) Art. 92, 2005
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O N H ARDY-H ILBERT I NTEGRAL I NEQUALITIES WITH S OME PARAMETERS
3
and
Z
∞
(1−αλ)+q(m−n)
1−q
y
(1.11)
∞
Z
0
f (x)
dx
α
(x + y α )λ
0
p
dy
Z
∞
x(1−αλ)+p(n−m) f p (x)dx,
e λ,α (m, n, p, q)
<H
0
where
1 1
Hλ,α (m, n, p, q) = B p
α
1 − mp
1 − mp
,λ −
α
α
B
1
q
1 − nq
1 − nq
,λ −
α
α
1 − nq
1 − nq
,λ −
α
α
and
e λ,α (m, n, p, q) = 1 B
H
αp
1 − mp
1 − mp
,λ −
α
α
B
p−1
.
Theorem 1.2. If p > 1, p1 + 1q = 1, α > 0, λ > 0, m, n ∈ R, such that 0 < 1 − mp < αλ,
mp + nq = 2 − αλ, and f (x) ≥ 0, g(y) ≥ 0, satisfy
Z ∞
(1.12)
0<
xn(p+q)−1 f p (x)dx < ∞,
0
Z
0<
(1.13)
∞
y m(p+q)−1 g q (y)dy < ∞,
0
then
Z
∞
∞
Z
(1.14)
0
0
f (x)g(x)
1
1 − mp
1 − mp
dxdy < B
,λ −
(xα + y α )λ
α
α
α
Z ∞
p1 Z ∞
1q
×
xn(p+q)−1 f p (x)dx
y m(p+q)−1 g q (y)dy ;
0
0
p
f (x)
(1.15)
y
dx dy
(xα + y α )λ
0
0
Z ∞
1 p 1 − mp
1 − mp
< pB
,λ −
xn(p+q)−1 f p (x)dx,
α
α
α
0
where the constant factors α1 B 1−mp
, λ − 1−mp
in (1.14) and α1p B p 1−mp
, λ − 1−mp
in (1.15)
α
α
α
α
are the best possible.
Z
∞
m(p+q)−1
1−q
Z
∞
2. W EIGHT F UNCTION AND L EMMAS
The weight function is defined as follows
Z ∞
1
yn
ωλ,α (m, n, y) =
·
dx,
(xα + y α )λ xm
0
y ∈ (0, +∞).
Lemma 2.1. If α > 0, λ > 0, m ∈ R, 0 < 1 − m < αλ, then
1 (1−αλ)+(n−m)
1−m
1−m
(2.1)
ωλ,α (m, n, y) = y
B
,λ −
.
α
α
α
J. Inequal. Pure and Appl. Math., 6(4) Art. 92, 2005
http://jipam.vu.edu.au/
4
YONG H ONG
Proof. Setting t =
xα
,
yα
then
Z
1−m
1 ∞
1
ωλ,α (m, n, y) =
y (1−αλ)+(n−m) t α −1 dt
λ
α 0 (1 + t)
Z
1−m
1 (1−αλ)+(n−m) ∞
1
t α −1 dt
= y
λ
α
(1 + t)
0
1 (1−αλ)+(n−m)
1−m
1−m
B
,λ −
.
= y
α
α
α
Hence (2.1) is valid. The lemma is proved.
Lemma 2.2. If α > 0, λ > 0, β < 1, a ∈ R, then
Z
∞
(2.2)
1
1
x1+ε
Z
0
1
xα
1−β
1
t α −1−aε dtdx = O(1),
λ
(1 + t)
Proof. Since (1 − β)/α > 0, for ε small enough, such that
Z
∞
1
1
x1+ε
Z
0
1
xα
1−β
α
(ε → 0+ ).
− aε > 0, then
Z 1
1 xα ( 1−β
t α −aε)−1 dtdx
x
1
0
Z ∞
1
=
xβ+aεα−2 dx
1 − β − aεα 1
1
=
.
(1 − β − aεα)2
1−β
1
t α −1−aε dtdx <
λ
(1 + t)
Z
∞
Hence (2.2) is valid.
3. P ROOFS OF THE T HEOREMS
Proof of Theorem 1.1. By Hölder’s inequality and Lemma 2.1, we have
Z ∞Z ∞
f (x)g(y)
G=
dxdy
(xα + y α )λ
0
0
Z ∞Z ∞
f (x)
xn
g(y)
ym
=
dxdy
(xα + y α )λ/p y m (xα + y α )λ/q xn
0
0
Z ∞ Z ∞
1 Z ∞ Z ∞
1q
f p (x) xnp p
g q (x) y mq
(3.1)
≤
dxdy
,
(xα + y α )λ y mp
(xα + y α )λ xnq
0
0
0
0
according to the condition of taking equality in Hölder’s inequality, if (3.1) takes equality, then
there exists a constant C, such that
f p (x) xnp
g q (y) y mq
≡ C, a.e. (x, y) ∈ (0, +∞) × (0, +∞)
(xα + y α )λ y mp
(xα + y α )λ xnq
it follows that
f p (x)xn(p+q) ≡ Cg q (y)y m(p+q) ≡ C1 (constant),
J. Inequal. Pure and Appl. Math., 6(4) Art. 92, 2005
a.e. (x, y) ∈ (0, +∞) × (0, +∞)
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O N H ARDY-H ILBERT I NTEGRAL I NEQUALITIES WITH S OME PARAMETERS
hence
Z
∞
x
∞
Z
(1−αλ)+p(n−m) p
x(1−αλ)+n(p+q)−nq−mp f p (x)dx
f (x)dx =
0
5
0
∞
Z
x(1−αλ)−np−mq dx
= C1
0
1
Z
= C1
x
(1−αλ)−np−mq
Z
∞
dx + C1
0
x(1−αλ)−np−mq dx
1
= +∞,
which contradicts (1.8) and (1.9). Hence, by (3.1), we have
Z ∞ Z ∞
p1
1
xnp
p
G<
dy f (x)dx
(xα + y α )λ y mp
0
0
Z ∞ Z ∞
1q
1
y mq
q
×
dy g (y)dy
(xα + y α )λ xnq
0
0
Z ∞
p1
1
1 − mp
1 − mp
p
(1−αλ)+p(n−m)
=
x
B
,λ −
f (x)dx
α 0
α
α
Z ∞
1q
1
1 − nq
1 − nq
(1−αλ)+q(m−n)
q
×
y
B
,λ −
g (y)dy
α 0
α
α
1q
Z ∞
p1 Z ∞
= Hλ,α (m, n, p, q)
x(1−αλ)+p(n−m) f p (x)dx
y (1−αλ)+q(m−n) g q (y)dy .
0
0
Hence (1.10) is vaild.
Let β = [(1 − αλ) + q(m − n)]/(1 − q), and
Z ∞
pq
f (x)
β
ge(y) = y
dx .
(xα + y α )λ
0
By (1.10), we have
Z ∞
y (1−αλ)+q(m−n) geq (y)dy
0
Z ∞
=
y β(1−q) geq (y)dy
p
Z0 ∞ Z ∞
f (x)
β
=
y
dx dy
(xα + y α )λ
0
0
pq Z ∞
Z ∞ Z ∞
f (x)
f (x)
β
=
y
dx
dx dy
(xα + y α )λ
(xα + y α )λ
0
0
0
Z ∞Z ∞
f (x)e
g (y)
=
dxdy
(xα + y α )λ
0
0
Z ∞
p1
(1−αλ)+p(n−m) p
< Hλ,α (m, n, p, q)
x
f (x)dx
0
Z
×
y
0
J. Inequal. Pure and Appl. Math., 6(4) Art. 92, 2005
∞
1q
ge (y)dy| ,
(1−αλ)+q(m−n) q
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6
YONG H ONG
It follows that
Z
Z ∞
(1−αλ)+q(m−n)
1−q
y
0
∞
0
f (x)
dx
α
(x + y α )λ
p
dy
Z
e λ,α (m, n, p, q)
<H
∞
x(1−αλ)+p(n−m) f p (x)dx.
0
Hence, (1.11) is valid.
Proof of Theorem 1.2. Since 0 < 1 − mp < αλ, mp + nq = 2 − αλ, then
0 < 1 − nq < αλ,
(1 − αλ) + p(n − m) = n(p + q) − 1,
(1 − αλ) + q(m − n) = m(p + q) − 1,
1 − mp
1 − nq
=λ−
.
α
α
By Theorem 1.1, (1.14) and (1.15) are valid.
For ε > 0, setting
( [−n(p+q)−ε]/p
x
, x ≥ 1;
f0 (x) =
0,
0 ≤ x < 1,
and
( [−m(p+q)−ε]/q
y
, y ≥ 1;
g0 (y) =
0,
0 ≤ y < 1.
We have
Z ∞
Z ∞
1
n(p+q)−1 p
(3.2)
0<
x
f0 (x)dx =
x−1−ε dx = < ∞,
ε
0
1
Z
0<
(3.3)
∞
y m(p+q)−1 g0q (y)dy
0
Z
0
∞
Z
Z
=
1
∞
y −1−ε dy =
1
< ∞.
ε
∞
f0 (x)g0 (y)
dxdy
(xα + y α )λ
0
Z ∞Z ∞
m(p+q)+ε
n(p+q)+ε
1
−
−
p
q
=
y
dxdy
x
α + y α )λ
(x
1
1
Z ∞
Z ∞
m(p+q)+ε
n(p+q)+ε
1
−
p
y − q dydx
=
x
α
α
λ
(x + y )
1
Z ∞
Z ∞1
1−mp
ε
1
1
1
=
t α −1− qα dtdx
1+ε
λ
1
α 1 x
(1 + t)
α
Z ∞
Zx ∞
1−mt
ε
1
1
1
=
t α −1− qα dtdx
1+ε
λ
α 1 x
(1 + t)
0
#
Z ∞
Z 1α
x
1−mt
ε
1
1
−
t α −1− qα dtdx .
1+ε
λ
x
(1
+
t)
1
0
By Lemma 2.2, when ε → 0, we have
Z ∞
Z 1α
x
1−mp
ε
1
1
t α −1− qα dtdx = O(1).
1+ε
λ
x
(1 + t)
1
0
J. Inequal. Pure and Appl. Math., 6(4) Art. 92, 2005
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O N H ARDY-H ILBERT I NTEGRAL I NEQUALITIES WITH S OME PARAMETERS
7
Since
Z
∞
0
we have
Z ∞Z
1−mp
ε
1
−1− qα
α
t
dt = B
λ
(1 + t)
1 − mp
1 − mp
,λ −
α
α
+ o(1),
∞
1 1
1 − mp
1 − mp
f0 (x)g0 (y)
dxdy =
B
,λ −
+ o(1) − O(1)
(xα + y α )λ
α ε
α
α
0
0
1 − mp
1 − mp
1 1
=
B
,λ −
− o(1)
ε α
α
α
1 − mp
1 − mp
1
(3.4)
=
B
,λ −
(1 − o(1)).
εα
α
α
1−mp
If the constant α1 B 1−mp
,
λ
−
in (1.14) is not the best possible, then there exists a K
α
α
<
1−mp
1−mp
1−mp
1−mp
1
1
B α , λ − α , such that (1.14) still is valid when we replace α B α , λ − α
by
α
K. By (3.2), (3.3) and (3.4), we find
1
1 − mp
1 − mp
B
,λ −
(1 − o(1))
εα
α
α
Z ∞
p1 Z ∞
1q
1
p
q
<K
xn(p+q)−1 f0 (x)dx
=K .
y m(p+q)−1 g0 (y)dy
ε
0
0
1
For ε → 0+ , we have
B 1−mp
, λ − 1−mp
≤ K, which
α
α
α
contradicts the fact that K <
1−mp
1−mp
1−mp
1−mp
1
1
B α , λ − α . It follows that α B α , λ − α
in (1.14) is the best
α
possible.
1−mp
Since (1.14) is equivalent to (1.15), then the constant α1p B p 1−mp
,
λ
−
in (1.15) is the
α
α
best possible. The theorem is proved.
4. S OME C OROLLARIES
When we take the appropriate parameters, many new inequalities can be obtained as follows:
Corollary 4.1. If p1 + 1q = 1 (p > 1), α > 0, λ > 0, f ≥ 0, g ≥ 0, and x(1−αλ)(p−1)/p f (x) ∈
Lp (0, +∞), x(1−αλ)(q−1)/q g(x) ∈ Lq (0, +∞), then
Z ∞Z ∞
f (x)g(y)
(4.1)
dxdy
(xα + y α )λ
0
0
p1 Z ∞
1q
Γ λp Γ λq Z ∞
<
x(1−αλ)(p−1) f p (x)dx
x(1−αλ)(q−1) g q (x)dx ;
αΓ(λ)
0
0
Z
(4.2)
0
∞
y α−1
Z
∞
0
where the constants Γ
are the best possible.
f (x)
dx
+ y α )λ
(xα
p
 p
Z ∞
Γ λp Γ λq


dy <
x(1−αλ)(p−1) f p (x)dx,
αΓ(λ)
0
h .
ip
.
λ
λ
λ
λ
Γ
(αΓ(λ))
in
(4.1)
and
Γ
Γ
(αΓ(λ))
in (4.2)
p
q
p
q
Proof. If we take m = p1 − αλ
, n = 1q − αλ
in Theorem 1.2, (4.1) and (4.2) can be obtained. p2
q2
J. Inequal. Pure and Appl. Math., 6(4) Art. 92, 2005
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8
YONG H ONG
Corollary 4.2. If p1 + 1q = 1 (p > 1), λ > 0, f ≥ 0, g ≥ 0 and x(1−λ)(p−1)/p f (x) ∈ Lp (0, +∞),
x(1−λ)(q−1)/q g(x) ∈ Lq (0, +∞), then
Z ∞Z ∞
f (x)g(y)
(4.3)
dxdy
(x + y)λ
0
0
1q
p1 Z ∞
Γ λp Γ λq Z ∞
(1−λ)(q−1) q
(1−λ)(p−1) p
<
x
g (x)dx ,
x
f (x)dx
Γ(λ)
0
0
Z
∞
Z
∞
(4.4)
0
0
f (x)
dx
(x + y)λ
p
 p
Z ∞
Γ λp Γ λq

dy < 
x(1−λ)(p−1) f p (x)dx,
Γ(λ)
0
.
h .
ip
λ
λ
λ
λ
Γ
Γ(λ)
in
(4.3)
and
Γ
Γ
Γ(λ)
in (4.4) are the best possible.
p
q
p
q
where Γ
Proof. If we take α = 1 in Corollary 4.1, (4.3) and (4.4) can be obtained.
Corollary 4.3. If p1 + 1q = 1 (p > 1), λ > 0, p + λ − 2 > 0, q + λ − 2 > 0, f ≥ 0, g ≥ 0, and
x(1−λ)/p f (x) ∈ Lp (0, +∞), x(1−λ)/q g(x) ∈ Lq (0, +∞), then
Z ∞Z ∞
f (x)g(y)
(4.5)
dxdy
(x + y)λ
0
0
1q
Z ∞
p1 Z ∞
p+λ−2 q+λ−2
1−λ q
1−λ p
<B
,
x f (x)
x g (x)dx ,
p
q
0
0
p
Z ∞
f (x)
p+λ−2 q+λ−2
p
(4.6)
y
dx dy < B
,
x1−λ f p (x)dx,
λ
(x
+
y)
p
q
0
0
0
where B p+λ−2
, q+λ−2
in (4.5) and B p p+λ−2
, q+λ−2
in (4.6) are the best possible.
p
q
p
q
∞
Z
1−λ
1−q
∞
Z
Proof. If we take α = 1, m = n =
2−λ
pq
in Theorem 1.2, (4.5) and (4.6) can be obtained.
Corollary 4.4. If p1 + 1q = 1 (p > 1), α > 0, f ≥ 0, g ≥ 0, and x(1−α)/p f (x) ∈ Lp (0, +∞),
x(1−α)/q g(x) ∈ Lq (0, +∞), then
Z ∞Z ∞
f (x)g(x)
(4.7)
dxdy
xα + y α
0
0
Z ∞
p Z ∞
q
π
1−α p
1−α q
1 x f (x)dx
x g (x)dx ,
<
1
π
π
0
0
sin q qα
α sin p pα
Z
∞
y
(4.8)
0
1−α
1−q
Z
0
∞
f (x)
dx
α
x + yα
Proof. If we take λ = 1, m = n =
J. Inequal. Pure and Appl. Math., 6(4) Art. 92, 2005
p
dy < 
1
,
pq
p

α sin
1
p
π
π
pα
Z
sin
1
q

π
qα
∞
x1−α f p (x)dx.
0
in Theorem 1.1, (4.7) and (4.8) can be obtained.
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O N H ARDY-H ILBERT I NTEGRAL I NEQUALITIES WITH S OME PARAMETERS
9
Corollary 4.5. If p1 + 1q = 1 (p > 1), α > 0, f ≥ 0, g ≥ 0, and f (x) ∈ Lp (0, +∞),
g(x) ∈ Lq (0, +∞), then
1
1
Z ∞
p1 Z ∞
1q
Z ∞Z ∞
Γ qα
Γ
pα
f (x)g(x)
p
q
(4.9)
f (x)dx
g (x)dx ,
1 dxdy <
αΓ α1
(xα + y α ) α
0
0
0
0
Z
∞
Z
(4.10)
∞
!p
f (x)
1
0
0
(xα + y α ) α
.
1
1
where Γ pα
Γ qα
αΓ
best possible.
1
α
dx
 p
1
1
Z ∞
Γ pα
Γ qα


dy <
f p (x)dx,
αΓ α1
0
h .
1
1
in (4.9) and Γ pα
Γ qα
αΓ
Proof. If we take λ = α1 , m = n =
1
pq
1
α
ip
in (4.10) are the
in Theorem 1.2, (4.9) and (4.10) can be obtained.
Remark 4.6. (4.1) and (4.2) are respectively generalizations of (1.6) and (1.7). For α = 1 in
(4.1) and (4.2), (1.6) and (1.7) can be obtained.
Remark 4.7. (4.3) and (4.4) are respectively generalizations of (1.1) and (1.2) .
Remark 4.8. (4.5) is the result of [1] and [2]. (4.6) is a new inequality.
Remark 4.9. (4.7) is the result of [3]. (1.7) is a new inequality.
Remark 4.10. (4.9) is a generalization of (1.5). (4.10) is a new inequality.
For other appropriate values of parameters taken in Theorems 1.1 and 1.2, many new inequalities and the inequalities of [6] – [13] can yet be obtained.
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