Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 3, Article 113, 2006
BEST GENERALIZATION OF A HILBERT TYPE INEQUALITY
BAOJU SUN
Z HEJIANG WATER C ONSERVANCY & H YDROPOWER C OLLEGE
H ANGZHOU , Z HEJIANG 310018
P EOPLE ’ S R EPUBLIC O F C HINA
sunbj@mail.zjwchc.com
Received 05 January, 2006; accepted 10 April, 2006
Communicated by B. Yang
A BSTRACT. By introducing a parameter λ , we have given generalization of Hilbert’s type integral inequality with a best possible constant factor. Also its equivalent form is considered, and
the generalized formula corresponding to the double series inequalities are built.
Key words and phrases: Hilbert’s type integral inequality; Weight coefficient; Weight function; Hölder’s inequality.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
R∞
R∞
If p > 1, p1 + 1q = 1, f, g ≥ 0, satisfy 0 < 0 f p (t)dt < ∞ and 0 < 0 g q (t)dt < ∞, then
1q
Z ∞
p1 Z ∞
Z ∞Z ∞
f (x)g(y)
π
p
q
g (t)dt ,
dxdy <
f (t)dt
(1.1)
x+y
sin( πp ) 0
0
0
0
where the constant factor π/(sin π/p) is the best possible. Inequality (1.1) is called HardyHilbert’s inequality (see [1]) and is important in analysis and applications (cf. Mitrinović et al.
[2]). Recently, Yang gave an extension of (1.1) as (see [4, 5]):
Z ∞Z ∞
f (x)g(y)
(1.2)
dxdy
(x + y)λ
0
0
Z ∞
p1 Z ∞
1q
p+λ−2 q+λ−2
1−λ p
1−λ q
<B
,
t f (t)dt
t g (t)dt ,
p
q
0
0
q+λ−2
where the constant factor B p+λ−2
,
(λ > 2 − min{p, q}) is the best possible, and
p
q
B(u, v) is the β function. Hardy et al. [1] gave an inequality similar to (1.1) as:
1q
Z ∞
p1 Z ∞
Z ∞Z ∞
f (x)g(y)
q
p
g (t)dt ,
(1.3)
dxdy < pq
f (t)dt
max{x, y}
0
0
0
0
ISSN (electronic): 1443-5756
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007-06
2
BAOJU S UN
where the constant factor pq is the best possible. The double series inequality is:
∞
X
am b n
< pq
apn
max{m,
n}
n=1
n=1 m=1
∞ X
∞
X
(1.4)
! p1
∞
X
! 1q
bqm
,
m=1
where the constant factor pq is the best possible. In particular, if p = q = 2, one has the Hilbert
type integral inequality:
∞
Z
Z
∞
(1.5)
0
0
f (x)g(y)
dxdy < 4
max{x, y}
∞
Z
Z
2
12
g (t)dt .
∞
2
f (t)dt
0
0
Recently, Kuang gave an extension of (1.4) as (see [3]):
∞ X
∞
X
am b n
<
max{m,
n}
n=1 m=1
(1.6)
∞
X
! p1
∞
X
[pq − G(p, n)]apn
n=1
! 1q
[pq − G(q, n)]bpm
,
m=1
where G(r, n) = r+1/3r−4/3
> 0 (r = p, q). Yang and Debnath have also considered other
(2n+1)1/r
Hilbert type integral inequalities in [6].
The main objective of this paper
build a new inequality with a best constant factor,
R ∞ R ∞is to
f (x)g(y)
related to the double integral 0 0 max{xλ ,yλ } dxdy, which improves inequality (1.5). The
equivalent form and the corresponding double series form are considered.
2. M AIN R ESULTS
Theorem 2.1. If λ > 0, p > 1, p1 + 1q = 1, f, g ≥ 0 such that 0 <
R∞
0 < 0 tq−1−λ g q (t)dt < ∞, then one has
∞
Z
Z
∞
(2.1)
0
0
f (x)g(y)
pq
dxdy <
λ
λ
max{x , y }
λ
Z
R∞
0
p1 Z
f (t)dt
∞
∞
p−1−λ p
t
0
tp−1−λ f p (t)dt < ∞,
1q
g (t)dt
q−1−λ q
t
0
and
p Z ∞
f (x)
pq p
(2.2)
y
dx
<
xp−1−λ f p (x)dx,
λ, yλ}
max{x
λ
0
0
0
p
where the constant factors pq
, pq
are the best possible. Inequality (2.1) is equivalent to (2.2).
λ
λ
In particular, for λ = 1, (2.1) and (2.2) respectively reduce to the following two equivalent
inequalities:
Z
Z
∞
λ(p−1)−1
∞
Z
∞
(2.3)
0
0
Z
∞
f (x)g(y)
dxdy < pq
max{x, y}
∞
Z
p1 Z
f (t)dt
p−2 p
t
0
∞
1q
f (t)dt
q−2 q
t
0
and
Z
∞
y
(2.4)
0
p−2
Z
0
∞
f (x)
dx
max{x, y}
J. Inequal. Pure and Appl. Math., 7(3) Art. 113, 2006
p
p
Z
dy < (pq)
∞
xp−2 f p (x)dx.
0
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B EST G ENERALIZATION OF A H ILBERT T YPE I NEQUALITY
3
Proof. By Hölder’s inequality, one has
Z
∞
Z
(2.5)
0
0
∞
f (x)g(y)
dxdy =
max{xλ , y λ }
Z
∞
Z
∞
"
y pqλ − p1
f (x)
1
p
1
1
xq−p
x
(max{xλ , y λ })
#
pqλ − 1q
1
1
x
g(y)
×
y p − q dxdy
1
λ
λ
y
q
(max{x , y })
Z ∞ Z ∞
p1
y λq −1 p
f p (x)
−1
≤
x q dxdy
max{xλ , y λ } x
0
0
) 1q
(Z Z
λp −1
∞
∞
q
q
x
g (y)
.
y p −1 dxdy
×
λ
λ
max{x , y } y
0
0
0
0
Equality holds in (2.5) if there are two constants A, B, such that A2 + B 2 6= 0 and
y λq −1 p
f p (x)
g q (y)
−1
q
A
x
=B
max{xλ , y λ } x
max{xλ , y λ }
λp −1
q
x
y p −1
y
p−λ p
a.e. in (0, ∞) × (0, ∞), or Ax
f (x) = By q−λ g q (y) =constant a.e. in (0, ∞) × (0, ∞), this
R ∞ p−1−λ
contradicts the fact that 0 < 0 t
f p (t)dt < ∞. Thus the inequality (2.5) is strict.
Define the weight function wλ (r, t) as:
Z ∞
u λr −1
1
λ−1
wλ (r, t) = t
du,
r = p, q; t ∈ (0, ∞).
max{tλ , uλ } t
0
By computing, one has:
wλ (p, t) =
(2.6)
and we obtain
Z ∞Z
0
0
∞
pq
= wλ (q, t),
λ
f (x)g(y)
dxdy
max{xλ , y λ }
Z ∞
p1 Z
p−1−λ p
<
wλ (q, t)t
f (t)dt
q−1−λ q
wλ (p, t)t
0
0
pq
=
λ
1q
f (t)dt
∞
Z
∞
p1 Z
f (t)dt
p−1−λ p
t
0
∞
1q
f (t)dt .
q−1−λ q
t
0
For 0 < ε < λ , setting fe(t), ge(t) as: t ∈ (0, 1), fe(t) = ge(t) = 0; t ∈ [1, ∞), fe(t) =
λ−p−ε
λ−q−ε
t p , ge(t) = t q
Z ∞Z ∞ e
f (x)e
g (y)
(2.7)
dxdy
max{xλ , y λ }
0
0
Z ∞
Z ∞
λ−q−ε
λ−p−ε
1
=
y q
x p dx dy
max{xλ , y λ }
1
1
Z ∞
Z y
Z ∞
Z ∞
λ−q−ε
λ−p−ε
λ−p−ε
λ−q−ε
1
1
y q
x p dx dy
x p dx dy +
=
y q
λ
xλ
1
y
1 y
1
J. Inequal. Pure and Appl. Math., 7(3) Art. 113, 2006
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BAOJU S UN
p
1
q
1
p
=
+
+ ·
λ − ε ε λ − λq − ε
ε λp − λ + ε
1
p
p
pq
=
+
−
.
ε λ − ε λp − λ + ε
(λ − ε)(λq − λ + ε)
On the other hand,
p1 Z
f (t)dt
∞
Z
p−1−λ ep
t
(2.8)
0
1q
1
ge (t)dt = .
ε
∞
q−1−λ q
t
0
If the constant factor pq
in (2.1) is not the best possible, then there exists a positive number k
λ
(with k < pq
),
such
that
(2.1)
is still valid if one replaces pq
by k. By (2.7) and (2.8), one has:
λ
λ
p
pqε
p
+
−
λ − ε λp − λ + ε (λ − ε)(λq − λ + ε)
Z ∞Z ∞ e e
f (x)f (y)
=ε
dxdy
max{xλ , y λ }
0
0
Z ∞
p1 Z ∞
1q
p−1−λ ep
q−1−λ q
< εk
t
f (t)dt
t
ge (t)dt
(2.9)
0
0
= k.
p
Setting ε → 0+ , then λp + (p−1)λ
≤ k or pq
≤ k. By this contradiction we can conclude that the
λ
pq
constant factor λ in (2.1) is the best possible.
Define the weight function g(y) as:
g(y) = y
λ(p−1)−1
Z
∞
0
f (x)
dx
max{xλ , y λ }
p−1
,
y ∈ (0, ∞).
By (2.1), one has:
Z
(2.10)
∞
0<
y
q−1−λ q
p
g (y)dy
Z ∞
0
p p
f (x)
=
y
dy
max{xλ , y λ }
0
0
Z ∞ Z ∞
p
f (x)g(y)
=
dxdy
max{xλ , y λ }
0
0
Z ∞
p−1
pq p Z ∞
p−1−λ p
q−1−λ q
≤
x
f (x)dx
y
g (y)dy
,
λ
0
0
Z
∞
λ(p−1)−1
Z
(2.11)
∞
y q−1−λ g q (y)dy
Z ∞
p
Z0 ∞
f (x)
λ(p−1)−1
=
y
dx dy
max{xλ , y λ }
0
0
pq p Z ∞
≤
xp−1−λ f p (x)dx < ∞.
λ
0
0<
J. Inequal. Pure and Appl. Math., 7(3) Art. 113, 2006
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B EST G ENERALIZATION OF A H ILBERT T YPE I NEQUALITY
5
Hence by using (2.1), (2.10) takes the form of strict inequality; so does (2.11). One then has
(2.2). On other hand, if (2.2) holds, by Hölder’s inequality, one has
Z ∞Z ∞
f (x)g(y)
(2.12)
dxdy
λ λ
0
0 max{x , y }
h
Z ∞
Z ∞
i
q−1−λ
λ+1−q
f (x)
q
q
g(y) dy
dx y
=
y
max{xλ , y λ }
0
0
1q
Z ∞
Z ∞
p p1 Z ∞
f (x)
λ(p−1)−1
q−1−λ q
≤
y
y
g (y)dy
max{xλ , y λ }
0
0
0
By (2.2), we have (2.1).
If the constant factor in(2.2) is not the best possible, we may show that the constant factor
in (2.1) is not the best possible by using (2.12). This is a contradiction. Hence the constant
factor in (2.2) is the best possible. Inequality (2.1) is equivalent to (2.2). Thus the theorem is
proved.
Remark 2.2. For p = q = 2, (2.3) reduces to (1.5); (2.1), (2.3) are generalizations of (1.5), but
(2.1) is not a generalization of (1.3).
Theorem 2.3. If p > 1, 1/p + 1/q = 1, 0 < λ ≤ min{p, q}, an ≥ 0, bn ≥ 0 such that
∞
∞
P
P
0<
np−1−λ apn < ∞, 0 <
nq−1−λ aqn < ∞, one has:
n=1
n=1
(2.13)
∞ X
∞
X
am b n
pq
<
λ
λ
max{m , n }
λ
n=1 m=1
∞
X
! p1
np−1−λ apn
n=1
∞
X
! 1q
nq−1−λ bqn
n=1
and
(2.14)
∞
X
n
∞
X
am
max{mλ , nλ }
m=1
λ(p−1)−1
n=1
!p
<
∞
pq p X
λ
np−1−λ apn ,
n=1
p
where the constant factors pq
, pq
are the best possible. Inequality (2.13) is equivalent to
λ
λ
(2.14).
In particular, for λ = 1, (2.13) and (2.14) respectively reduce to the following two equivalent
inequalities:
(2.15)
∞ X
∞
X
∞
X
am b n
< pq
np−2 apn
max{m,
n}
n=1 m=1
n=1
! p1
∞
X
! 1q
nq−2 bqn
n=1
and
(2.16)
∞
X
np−2
n=1
∞
X
am
max{m, n}
m=1
!p
< (pq)p
∞
X
np−2 apn .
n=1
Proof. Define the weight coefficient w
eλ (λ, n) as:
(2.17)
w
eλ (r, n) = n
λ−1
∞
X
m λr −1
1
,
max{mλ , nλ } n
m=1
J. Inequal. Pure and Appl. Math., 7(3) Art. 113, 2006
r = p, q; n ∈ N.
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BAOJU S UN
Since 0 < λ ≤ min{p, q}, we have:
∞ Z
X
m
u λr −1
1
du
λ , nλ } n
max{u
m−1
m=1
Z ∞
u λr −1
1
λ−1
du
=n
max{uλ , nλ } n
0
pq
= wλ (r, n) = , r = p, q
λ
By Hölder’s inequality and (2.17), following the method of proof in Theorem 2.1, one has:
! p1
! 1q
∞
∞
∞ X
∞
X
X
X
am b n
<
w
eλ (q, n)np−1−λ apn
w
eλ (p, n)nq−1−λ apn
λ , nλ }
max{m
m=1
n=1 m=1
n=1
(2.18)
w
eλ (r, n) < n
λ−1
λ−p−ε
λ−q−ε
By (2.18) we have (2.13), for 0 < ε < λ, setting e
an = n p , ebn = n q , n ∈ N. Since
0 < λ ≤ min{p, q}, by (2.9), we get:
Z ∞Z ∞ e
∞ X
∞
X
e
amebn
f (x)e
g (y)
>
dxdy
λ
λ
max{m , n }
max{xλ , y λ }
1
1
n=1 m=1
1
p
p
pq
=
+
−
.
ε λ − ε λp − λ + ε
(λ − ε)(λq − λ + ε)
On other hand,
! p1
! 1q
∞
∞
∞
X
X
X
1
1
p−1−λ p
q−1−λeq
n
e
an
<1+ .
n
bn
=
1+ε
n
ε
n=1
m=1
n=1
By using the above inequalities and the method of proof in Theorem 2.1, we may show that the
constant factor in (2.13) is the best possible.
Setting bn as:
!p−1
∞
X
a
m
bn = nλ(p−1)−1
,
max{mλ , nλ }
m=1
we obtain:
∞
X
nq−1−λ bqn =
n=1
∞
X
nλ(p−1)−1
n=1
=
∞ X
∞
X
m=1 n=1
∞
X
am
max{mλ , nλ }
m=1
!p
am b n
.
max{mλ , nλ }
By (2.13) and using the same method of Theorem 2.1, we have (2.14). We may show that the
constant factor in (2.14)is the best possible, and inequality (2.13) is equivalent to (2.14).
R EFERENCES
[1] G.H. HARDY, J.E. LITTLEWOOD AND G. POLYA, Inequalities, Cambridge University Press Cambridge, 1952.
[2] D.S. MITRINOVIĆ, J.E. PEC̆ARIĆ AND A.M. FINK, Inequalities Involving Functions and Their
Integrals and Derivatives, Kluwer Academic Publishers, Boston, 1991.
[3] J. KUANG AND L. DEBNATH, On new generalizations of Hilbert’s inequality and their applications, Math. Anal. Appl., 245(1) (2000), 248–265.
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B EST G ENERALIZATION OF A H ILBERT T YPE I NEQUALITY
7
[4] B. YANG, On a generalization of Hardy Hilbert’s integral inequality with a best value, Chinese Ann.
of Math., A21(4) (2000), 401–408.
[5] B. YANG, On Hardy Hilbert’s integral inequality, J. Math. Anal. Appl., 261 (2001), 295–306.
[6] B. YANG AND L. DEBNATH, On a new strengthened Hardy-Hilbert’s inequality, Internat. J. Math.
Sci., 21(1) (1998), 403–408.
J. Inequal. Pure and Appl. Math., 7(3) Art. 113, 2006
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