ON SOME WEIGHTED MIXED NORM HARDY-TYPE
INTEGRAL INEQUALITIES
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. IMORU AND A.G. ADEAGBO-SHEIKH
Department of Mathematics
Obafemi Awolowo University
Ile-Ife, Nigeria
EMail: cimoru@oauife.edu.ng
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
adesheikh2000@yahoo.co.uk
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Contents
Received:
07 May, 2007
Accepted:
24 August, 2007
Communicated by:
B. Opić
2000 AMS Sub. Class.:
26D15.
Key words:
Hardy-type inequality, Weighted norm.
Abstract:
In this paper, we establish a weighted mixed norm integral inequality of Hardy’s type.
This inequality features a free constant term and extends earlier results on weighted
norm Hardy-type inequalities. It contains, as special cases, some earlier inequalities
established by the authors and also provides an improvement over them.
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Contents
1
Introduction
3
2
Main Result
5
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
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1.
Introduction
In a recent paper [2], the authors proved the following result.
Theorem 1.1. Let g be continuous and non-decreasing on [a, b], 0 ≤ a ≤ b ≤ ∞
with g(x) > 0, x > 0, r 6= 1 and let f (x) be non-negative Rand Lebesgue-Stieltjes
x
integrable with respect to g(x) on [a, b]. Suppose Fa (x) = a f (t)dg(t), Fb (x) =
Rb
, r 6= 1. Then
f (t)dg(t) and δ = 1−r
p
x
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
Z
b
(1.1)
a
Z
1−p
g(x)δ−1 g(x)−δ − g(a)−δ
Fa (x)p dg(x) + K1 (p, δ, a, b)
p Z b
p
≤
g(x)δp−1 [g(x)f (x)]p dg(x),
r−1
a
b
δ−1
g(x)
(1.2)
a
1−p
g(x)−δ − g(b)−δ
Fb (x)p dg(x) + K2 (p, δ, a, b)
p Z b
p
≤
g(x)δp−1 [g(x)f (x)]p dg(x),
1−r
a
vol. 8, iss. 4, art. 101, 2007
r > 1,
Contents
r < 1,
where
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1−p
p
K1 (p, δ, a, b) =
g(b)δ g(b)−δ − g(a)−δ
Fa (b)p ,
r−1
δ < 0, i.e. r > 1
and
K2 (p, δ, a, b) =
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1−p
p
g(a)δ g(a)−δ − g(b)−δ
Fb (a)p ,
1−r
δ > 0, i.e. r < 1.
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The above result generalizes Imoru [1] and therefore Shum [3]. The purpose
of the present work is to obtain a weighted norm Hardy-type inequality involving
mixed norms which contains the above result as a special case and also provides an
improvement over it.
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
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2.
Main Result
The main result of this paper is the following theorem:
Theorem 2.1. Let g be a continuous function which is non-decreasing on [a, b], 0 ≤
a ≤ b < ∞, with g(x) > 0 for x > 0. Suppose that q ≥ p ≥ 1 and f (x) is
non-negative and Lebesgue-Stieltjes integrable with respect to g(x) on [a, b]. Let
Z x
Z x
(2.1)
Fa (x) =
f (t)dg(t), θa (x) =
g(t)(p−1)(1+δ) f (t)p dg(t),
a
Z
Fb (x) =
(2.2)
a
b
and δ =
1−r
,r
p
Z
b
(2.3)
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
b
Z
g(t)(p−1)(1+δ) f (t)p dg(t)
f (t)dg(t), θb (x) =
x
Weighted Mixed Norm Hardy-Type
Integral Inequalities
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x
Contents
6= 1. Then if r > 1, i.e. δ < 0,
1q
q
δq
(p−1)
g(x) p −1 g(x)−δ − g(a)−δ p
Faq (x)dg(x) + A1 (p, q, a, b, δ)
a
Z
b
≤ C1 (p, q, δ)
p1
p
δp−1
g(x)
[g(x)f (x)] dg(x) ,
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and for r < 1, i.e. δ > 0,
Z
(2.4)
b
δq
g(x) p −1 g(x)−δ − g(b)−δ
q
(p−1)
p
1q
Fbq (x)dg(x) + A2 (p, q, a, b, δ)
a
Z
≤ C2 (p, q, δ)
b
δp−1
g(x)
a
p1
[g(x)f (x)] dg(x) ,
p
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where
q
δq
q
p
(−δ) p (1−p)−1 g(b) p θa (b) p ,
q
1q
q
p
(1−p)−1
C1 (p, q, δ) = (−δ) p
,
q
δq
q
p q
A2 (p, q, a, b, δ) = (δ) p (1−p)−1 g(a) p θb (a) p ,
q
1
p pq (1−p)−1 q
.
C2 (p, q, δ) = δ
q
A1 (p, q, a, b, δ) =
δ < 0,
δ>0
Proof. For the proof of Theorem 2.1 we will use the following adaptations of Jensen’s
inequality for convex functions,
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
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x
Z
1
x
Z
h(x, t) pq dλ(t) ≤
(2.5)
1− p1 Z
dλ(t)
a
a
x
p1
1
h(x, t) q dλ(t)
a
and
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Z
b
1
Z
h(x, t) pq dλ(t) ≤
(2.6)
x
b
1− p1 Z b
p1
1
dλ(t)
h(x, t) q dλ(t) ,
x
x
where h(x, t) ≥ 0 for x ≥ 0, t ≥ 0, λ is non-decreasing and q ≥ p ≥ 1.
Let
(2.7)
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h(x, t) = g(x)δq g(t)pq(1+δ) f (t)pq ,
q
q
dλ(t) = g(t)−(1+δ) dg(t),
∆q1 = (−δ) p (1−p) , if δ < 0 and ∆q2 = (δ) p (1−p) , if δ > 0.
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Using (2.7) in (2.5), we get
Z x
δ
p
g(x)
f (t)dg(t)
a
≤ (−δ)
1
(1−p)
p
1 (p−1)
δ
g(x)−δ − g(a)−δ p
g(x) p
x
Z
(p−1)(1+δ)
g(t)
p1
f (t) dg(t) .
p
a
Raising both sides of the above inequalities to power q and using (2.1), we obtain
δq
p
q
(p−1)
p
δq
p
q
p
g(x) Fa (x)q ≤ ∆q1 ga (x)
g(x) θa (x) ,
where ga (x) = g(x)−δ − g(a)−δ .
Integrating over (a, b) with respect to g(x)−1 dg(x) gives
Z b
Z b
q
δq
q
δq
q
−1
(1−p)
q
p
p
(2.8)
g(x)
ga (x)
Fa (x) dg(x) ≤ ∆1
g(x) p −1 θa (x) p dg(x) = J.
a
a
Now integrate the right side of (2.8) by parts to obtain
Z b
δq
q
q
J = ∆1
g(x) p −1 θa (x) p dg(x)
=
a
∆q1
δq
q
g(x) p θa (x) p |ba + (−δ −1 )∆q1
(δq/p)
Z b
δq
q
×
g(x) p g(x)(p−1)(1+δ) f (x)p θa (x) p −1 dg(x).
a
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
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However,
Z b
q
δq
−1
I=
g(x) p g(x)(p−1)(1+δ) f (x)p θap (x)dg(x)
a
b
Z
=
δq
p
(p−1)(1+δ)
g(x) g(x)
g(t)
pq −1
f (t) dg(t)
dg(x)
p
a
b
=
δp+p−1−δ
f (x)
a
Z
x
Z
p
x
Z
δp+p−1
p
δ
g(x)
f (x) g(x)
a
pq −1
δp+p−1−δ
p
dg(x).
g(t)
f (t) dg(t)
a
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
Since δ < 0, we have g(x)−δ ≥ g(t)−δ ∀t ∈ [a, x].
Consequently
Z x
pq −1
Z b
δp+p−1
p
δp+p−1
p
I≤
g(x)
f (t)
g(t)
f (t) dg(t)
dg(x)
a
a
Z b Z
x
δp+p−1
=
g(t)
a
p
=
q
pq −1
g(x)δp+p−1 f (t)p dg(x)
f (t) dg(t)
p
a
b
Z
δp−1
g(x)
pq
[f (x)g(x)] dg(x) .
p
a
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Thus (2.8) becomes
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Z
b
g(x)
(2.9)
a
δq
−1
p
q (1−p)
g(x)−δ − g(a)−δ p
Fa (x)q dg(x)
δq
q
p
+ ∆q1 (−δ −1 )g(b) p θa (b) p
q
p
≤ (−δ −1 )∆q1
q
Z
b
δp−1
g(x)
pq
[f (x)g(x)] dg(x) .
p
a
Taking the q th root of both sides yields assertion (2.3) of the theorem.
To prove (2.4), we start with inequality (2.6) and use (2.7) with (2.2) to obtain
δ
1
1
δ
1
g(x) p Fb (x) ≤ (−δ) p (1−p) gb (x) p (p−1) g(x) p θb (x) p
−1
1
(p−1)
p
1
(p−1)
p
δ
p
1
p
= (δ )
(−gb (x))
g(x) θb (x) ,
where gb (x) = g(b)−δ − g(x)−δ .
On rearranging and raising to power q and then integrating both sides over [a, b]
with respect to g(x)−1 dg(x), we obtain
Z b
q (1−p)
δq
(2.10)
g(x) p −1 g(x)−δ − g(b)−δ p
Fb (x)q dg(x)
a
Z b
δq
q
≤ ∆2
g(x) p −1 θb (x)dg(x).
a
We denote the right side of (2.10) by H, integrate it by parts and use the fact that for
δ > 0, g(x)δ ≤ g(t)δ ∀t ∈ [x, b] to obtain
Z b
δp
q
p q
q
b
−1
H ≤ ∆2 g(x) q θb (x) p |a + (δq/p) ∆2
g(x)δp−1 [f (x)g(x)]p dg(x).
δq
a
Using this in (2.10) we obtain
Z b
q (1−p)
q
q
δq
p
Fb (x)q dg(x) + δ −1 ∆q2 g(a) p θb (a) p
(2.11)
g(x) p −1 g(x)−δ − g(b)−δ p
q
a
Z b
pq
p −1 q
p
δp−1
≤ δ ∆2
g(x)
[f (x)g(x)] dg(x) .
q
a
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
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We take the q th root of both sides to obtain assertion (2.4) of the theorem.
Remark 1. Let p = q, and δ =
Z
(2.12)
a
b
1−r
p
< 0, i.e., r > 1, then (2.3) reduces to
p−1
g(x)δ−1 g(x)−δ − g(a)−δ
Fa (x)p dg(x) + A1 (p, p, a, b, δ)
Z b
p
δp−1
≤ C1 (p, p, δ)
g(x)
[f (x)g(x)] dg(x) ,
a
C.O. Imoru and A.G. Adeagbo-Sheikh
where
(2.13)
vol. 8, iss. 4, art. 101, 2007
A1 (p, p, a, b, δ) = (−δ)−p g(b)δ θa (b),
δ<0
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and
(2.14)
−p
C1 (p, p, δ) = (−δ)
p
=
r−1
p
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.
Now from (2.18) in [2] we have that, for δ < 0
1−p
(2.15)
g(b)δ θa (b) ≥ (−δ −1 )1−p g(b)δ g(b)−δ − g(a)−δ
Fa (b)p .
Thus, from (2.13) and (2.15), using notations in (1.1), we have
(2.16)
Weighted Mixed Norm Hardy-Type
Integral Inequalities
A1 (p, p, a, b, δ) = (−δ)−p g(b)δ θa (b)
1−p
≥ (−δ)−p (−δ −1 )1−p g(b)δ g(b)−δ − g(a)−δ
Fa (b)p
1−p
= (−δ)−1 g(b)δ g(b)−δ − g(a)−δ
Fa (b)p
1−p
p
g(b)δ g(b)−δ − g(a)−δ
Fa (b)p
=
r−1
= K1 (p, δ, a, b),
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i.e., A1 (p, p, a, b, δ) = K1 (p, δ, a, b) + B1 for some B1 ≥ 0.
Thus we can write (2.12), using (2.14), as
Z
(2.17)
a
b
p−1
g(x)δ−1 g(x)−δ − g(a)−δ
Fa (x)p dg(x) + K1 (p, δ, a, b) + B1
p Z b
p
p
δp−1
g(x)
[f (x)g(x)] dg(x) .
≤
r−1
a
So, when B1 = 0, (2.17) reduces to (1.1). When B1 6= 0, i.e., B1 > 0, (2.17) is an
improvement of (1.1). Similarly with notations in (1.2) and (2.4) in this paper we
use (2.19) in [2] to prove that
A2 (p, p, a, b, δ) = K2 (p, δ, a, b) + B2
for some B2 ≥ 0.
Thus, when p = q, (2.4) reduces to (1.2) if B2 = 0 and is an improvement of (1.2)
when B2 6= 0, i. e., when B2 > 0.
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
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References
[1] C.O. IMORU, On some integral inequalities related to Hardy’s, Canadian Math.
Bull., 20(3) (1977), 307–312.
[2] C.O. IMORU AND A.G. ADEAGBO-SHEIKH, On an integral inequality of the
Hardy-type, (accepted) Austral. J. Math. Anal. and Applics.
[3] D.T. SHUM, On integral inequalities related to Hardy’s, Canadian Math. Bull.,
14(2) (1971), 225–230.
Weighted Mixed Norm Hardy-Type
Integral Inequalities
C.O. Imoru and A.G. Adeagbo-Sheikh
vol. 8, iss. 4, art. 101, 2007
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