Journal of Inequalities in Pure and
Applied Mathematics
BEST GENERALIZATION OF A HILBERT TYPE INEQUALITY
BAOJU SUN
Zhejiang Water Conservancy & Hydropower College
Hangzhou, Zhejiang 310018
People’s Republic Of China.
volume 7, issue 3, article 113,
2006.
Received 05 January, 2006;
accepted 10 April, 2006.
Communicated by: B. Yang
EMail: sunbj@mail.zjwchc.com
Abstract
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Victoria University
ISSN (electronic): 1443-5756
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Abstract
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.
Best Generalization of a Hilbert
Type Inequality
2000 Mathematics Subject Classification: 26D15.
Key words: Hilbert’s type integral inequality; Weight coefficient; Weight function;
Hölder’s inequality.
Baoju Sun
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
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1.
Introduction
If p > 1, p1 + 1q = 1, f, g ≥ 0, satisfy 0 <
R∞ q
g (t)dt < ∞, then
0
∞
Z
∞
Z
(1.1)
0
0
f (x)g(y)
π
dxdy <
x+y
sin( πp )
Z
∞
R∞
0
f p (t)dt < ∞ and 0 <
p1 Z
f p (t)dt
0
∞
1q
g q (t)dt ,
0
where the constant factor π/(sin π/p) is the best possible. Inequality (1.1) is
called Hardy-Hilbert’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
(1.2)
0
0
Contents
Z
p1 Z
∞
1−λ p
t f (t)dt
1q
∞
1−λ q
t g (t)dt ,
p+λ−2 q+λ−2
,
p
q
0
0
q+λ−2
where the constant factor B p+λ−2
,
(λ > 2 − min{p, q}) is the best
p
q
possible, and B(u, v) is the β function. Hardy et al. [1] gave an inequality
similar to (1.1) as:
<B
Z
∞
Z
(1.3)
0
0
∞
Baoju Sun
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f (x)g(y)
dxdy
(x + y)λ
Best Generalization of a Hilbert
Type Inequality
f (x)g(y)
dxdy < pq
max{x, y}
Z
0
∞
p1 Z
f (t)dt
p
0
∞
1q
g (t)dt ,
q
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where the constant factor pq is the best possible. The double series inequality
is:
! p1
! 1q
∞
∞
∞ X
∞
X
X
X
am b n
(1.4)
< pq
apn
bqm
,
max{m,
n}
n=1
m=1
n=1 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
f 2 (t)dt
0
Z
∞
Best Generalization of a Hilbert
Type Inequality
21
2
g (t)dt .
Baoju Sun
0
Recently, Kuang gave an extension of (1.4) as (see [3]):
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(1.6)
∞ X
∞
X
am b n
max{m, n}
n=1 m=1
<
∞
X
n=1
Contents
! p1
[pq − G(p, n)]apn
∞
X
! 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
(2n+1)1/r
considered other Hilbert type integral inequalities in [6].
The main objective of this paper is to buildR a new
inequality with a best
∞ R ∞ f (x)g(y)
constant factor, 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.
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2.
Main Results
Theorem 2.1. If λ > 0, p > 1, p1 + 1q = 1, f, g ≥ 0 such that 0 <
R ∞ p−1−λ p
R∞
t
f (t)dt < ∞, 0 < 0 tq−1−λ g q (t)dt < ∞, then one has
0
Z ∞Z ∞
f (x)g(y)
(2.1)
dxdy
max{xλ , y λ }
0
0
Z ∞
p1 Z ∞
1q
pq
p−1−λ p
q−1−λ q
t
f (t)dt
t
g (t)dt
<
λ
0
0
and
Baoju Sun
Z
∞
Z
p
∞
pq p Z
∞
f (x)
dx <
xp−1−λ f p (x)dx,
λ, yλ}
max{x
λ
0
0
0
pq
pq p
where the constant factors λ , λ are the best possible. Inequality (2.1) is
equivalent to (2.2). In particular, for λ = 1, (2.1) and (2.2) respectively reduce
to the following two equivalent inequalities:
Z ∞Z ∞
f (x)g(y)
(2.3)
dxdy
max{x, y}
0
0
1q
Z ∞
p1 Z ∞
q−2 q
p−2 p
< pq
t f (t)dt
t f (t)dt
(2.2)
Best Generalization of a Hilbert
Type Inequality
y λ(p−1)−1
0
0
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and
Z
(2.4)
0
∞
y p−2
Z
0
∞
f (x)
dx
max{x, y}
p
dy < (pq)p
Z
0
∞
xp−2 f p (x)dx.
J. Ineq. Pure and Appl. Math. 7(3) Art. 111, 2006
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Proof. By Hölder’s inequality, one has
Z ∞Z ∞
f (x)g(y)
(2.5)
dxdy
λ λ
0
0 max{x , y }
Z ∞Z ∞"
y pqλ − p1 1 1
f (x)
=
xq−p
1
0
0
(max{xλ , y λ }) p x
#
pqλ − 1q
1
1
g(y)
x
−
×
y p q dxdy
1
λ
λ
y
q
(max{x , y })
p1
Z ∞ Z ∞
y λq −1 p
f p (x)
−1
q
x dxdy
≤
max{xλ , y λ } x
0
0
) 1q
(Z Z
λp −1
∞
∞
q
x
g q (y)
.
×
y p −1 dxdy
λ, yλ}
max{x
y
0
0
2
2
Equality holds in (2.5) if there are two constants A, B, such that A + B 6= 0
and
λp −1
y λq −1 p
q
q
f p (x)
g
(y)
x
−1
−1
q
p
A
x
=
B
y
λ
λ
λ
λ
max{x , y } x
max{x , y } y
a.e. in (0, ∞) × (0, ∞), or Axp−λ f p (x) = By q−λRg q (y) =constant a.e. in
∞
(0, ∞) × (0, ∞), this contradicts the fact that 0 < 0 tp−1−λ 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
Best Generalization of a Hilbert
Type Inequality
Baoju Sun
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By computing, one has:
wλ (p, t) =
(2.6)
pq
= wλ (q, t),
λ
and we obtain
Z ∞Z ∞
f (x)g(y)
dxdy
λ λ
0
0 max{x , y }
p1 Z
Z ∞
p−1−λ p
<
wλ (q, t)t
f (t)dt
1q
f (t)dt
q−1−λ q
wλ (p, t)t
Best Generalization of a Hilbert
Type Inequality
0
0
pq
=
λ
∞
Z
0
∞
p1 Z
p−1−λ p
t
f (t)dt
∞
1q
f (t)dt .
Baoju Sun
q−1−λ q
t
0
Title Page
For 0 < ε < λ , setting fe(t), ge(t) as: t ∈ (0, 1), fe(t) = ge(t) = 0; t ∈ [1, ∞),
λ−p−ε
λ−q−ε
fe(t) = t p , ge(t) = t q
Z
∞
Z
(2.7)
0
0
∞
fe(x)e
g (y)
dxdy
max{xλ , y λ }
Z ∞
Z ∞
λ−p−ε
λ−q−ε
1
p
q
dx dy
=
y
x
max{xλ , y λ }
1
1
Z y
Z ∞
λ−q−ε
1 λ−p−ε
q
p
x
=
y
dx dy
λ
1
1 y
Z ∞
Z ∞
λ−q−ε
λ−p−ε
1
+
y q
x p dx dy
xλ
1
y
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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,
Z ∞
p1 Z
p−1−λ ep
(2.8)
t
f (t)dt
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
λ
positive number k (with k < pq
), such that (2.1) is still valid if one replaces pq
λ
λ
by k. By (2.7) and (2.8), one has:
p
p
pqε
(2.9)
+
−
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
0
0
= k.
p
Setting ε → 0+ , then λp + (p−1)λ
≤ k or pq
≤ k. By this contradiction we can
λ
pq
conclude that the constant factor λ in (2.1) is the best possible.
Define the weight function g(y) as:
Z ∞
p−1
f (x)
λ(p−1)−1
g(y) = y
dx
,
y ∈ (0, ∞).
max{xλ , y λ }
0
Best Generalization of a Hilbert
Type Inequality
Baoju Sun
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By (2.1), one has:
Z ∞
p
q−1−λ q
(2.10)
0<
y
g (y)dy
0
Z ∞
Z ∞
p p
f (x)
λ(p−1)−1
=
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
(2.11)
∞
y q−1−λ g q (y)dy
0
Z ∞
p
Z ∞
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<
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
max{xλ , y λ }
0
0
Best Generalization of a Hilbert
Type Inequality
Baoju Sun
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∞
Z
λ+1−q
q
y
∞
h
i
q−1−λ
f (x)
q
=
dx
y
g(y)
dy
max{xλ , y λ }
0
0
Z ∞
Z ∞
p p1 Z ∞
1q
f (x)
λ(p−1)−1
q−1−λ q
≤
y
y
g (y)dy
max{xλ , y λ }
0
0
0
Z
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 1. 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 2.2. If p > 1, 1/p + 1/q = 1, 0 < λ ≤ min{p, q}, an ≥ 0, bn ≥ 0
∞
∞
P
P
such that 0 <
np−1−λ apn < ∞, 0 <
nq−1−λ aqn < ∞, one has:
n=1
(2.13)
n=1
∞ 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
Best Generalization of a Hilbert
Type Inequality
Baoju Sun
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∞
X
∞
pq p X
am
(2.14)
n
<
np−1−λ apn ,
λ , nλ }
max{m
λ
n=1
m=1
n=1
p
where the constant factors pq
, pq
are the best possible. Inequality (2.13) is
λ
λ
equivalent to (2.14).
λ(p−1)−1
∞
X
!p
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In particular, for λ = 1, (2.13) and (2.14) respectively reduce to the following two equivalent inequalities:
(2.15)
∞ X
∞
X
am b n
< pq
max{m, n}
n=1 m=1
∞
X
! p1
np−2 apn
n=1
∞
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
Best Generalization of a Hilbert
Type Inequality
Baoju Sun
Proof. Define the weight coefficient w
eλ (λ, n) as:
(2.17) w
eλ (r, n) = n
λ−1
∞
X
m λr −1
1
,
λ , nλ }
max{m
n
m=1
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r = p, q; n ∈ N.
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Since 0 < λ ≤ min{p, q}, we have:
(2.18)
∞ Z
X
m
u λr −1
1
du
λ , nλ } n
max{u
m−1
m=1
Z ∞
u λr −1
1
λ−1
=n
du
max{uλ , nλ } n
0
pq
= wλ (r, n) = , r = p, q
λ
w
eλ (r, n) < n
λ−1
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By Hölder’s inequality and (2.17), following the method of proof in Theorem
2.1, one has:
∞ X
∞
X
∞
X
am b n
<
λ , nλ }
max{m
n=1 m=1
! p1
w
eλ (q, n)np−1−λ apn
∞
X
e
amebn
>
λ , nλ }
max{m
n=1 m=1
Z
∞Z
w
eλ (p, n)nq−1−λ apn
m=1
n=1
By (2.18) we have (2.13), for 0 < ε < λ, setting e
an = n
n ∈ N. Since 0 < λ ≤ min{p, q}, by (2.9), we get:
∞ X
∞
X
! 1q
λ−p−ε
p
, ebn = n
λ−q−ε
q
,
Best Generalization of a Hilbert
Type Inequality
∞
fe(x)e
g (y)
dxdy
max{xλ , y λ }
1
1
p
p
pq
1
+
−
.
=
ε λ − ε λp − λ + ε
(λ − ε)(λq − λ + ε)
Baoju Sun
Title Page
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On other hand,
∞
X
n=1
! p1
np−1−λe
apn
∞
X
m=1
! 1q
nq−1−λebqn
=
∞
X
n=1
1
n1+ε
1
<1+ .
ε
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
am
λ(p−1)−1
bn = n
,
max{mλ , nλ }
m=1
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we obtain:
∞
X
nq−1−λ bqn
=
n=1
=
∞
X
n
λ(p−1)−1
n=1
∞ X
∞
X
m=1 n=1
∞
X
am
max{mλ , nλ }
m=1
!p
am bn
.
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).
Best Generalization of a Hilbert
Type Inequality
Baoju Sun
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References
[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.
[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.
Best Generalization of a Hilbert
Type Inequality
Baoju Sun
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