1. Introduction
The classical hardy integral inequality reads:
Theorem 1 Let
be a non-negative p-integrable function defined on
, and
. Then,
is integrable over the interval
for each
and the following inequality:
(1)
holds, where
is the best possible constant (see [1]).
This inequality can be found in many standard books (see [2-7]). Inequality (1) has found much interest from a number of researchers and there are numerous new proofs, as well as, extensions, refinements and variants which is refer to as Hardy type inequalities.
In the recent paper [8], the author proved the following generalization which is an extension of [9].
Theorem 2 Let
,
and
be finite, non-negative measurable functions on
,
and ![](https://www.scirp.org/html/1-5300467\cfd8bfa3-8b92-4a53-a63e-9f2f4d417ee3.jpg)
with
such that
. Then, the following inequality holds:
(2)
where,
![](https://www.scirp.org/html/1-5300467\8700e1c6-527e-4c9a-bb65-051a98ce81b7.jpg)
and
![](https://www.scirp.org/html/1-5300467\1b22b8b7-e990-47de-9dd5-cd9d35751d6b.jpg)
[10] also proved the following integral inequality of Hardy-type mainly by Jensen’s Inequality:
Theorem 3 Let
be continuous and nondecreasing on
with
for
Let
and
be nonnegative and LebesgueStieltjes integrable with respect to
on ![](https://www.scirp.org/html/1-5300467\44fedbed-5ea8-48dd-8bea-d2f1ded2e763.jpg)
Suppose
is a real number such that
then
(3)
where,
![](https://www.scirp.org/html/1-5300467\0cf8e4d7-7b63-4a86-8f7e-5e1e32f59a99.jpg)
Other recent developments of the Hardy-type inequalities can be seen in the papers [11-16]. In this article, we point out some other Hardy-type inequalities which will complement the above results (2) and (3).
2. Main Results
The following lemma is of particular interest (see also [8]).
Lemma. Let
,
,
, and let
be a non-negative measurable function such that
. Then the following inequality holds:
(4)
Proof
Let
![](https://www.scirp.org/html/1-5300467\7e971aeb-d267-4793-987e-01927a6e4503.jpg)
then,
![](https://www.scirp.org/html/1-5300467\f7e6bdda-be65-471e-a8e4-d13aaf2f3cea.jpg)
by Holder’s inequality, we have,
![](https://www.scirp.org/html/1-5300467\c781d38a-6eb3-4942-8775-25df7378ada9.jpg)
We need to show that there exists
such that for any
, equality in (4) does not hold. If otherwise, there exist a decreasing sequence
in
,
such that for
the inequality (4), written
, becomes an equality. Then, to every
there correspond real constants
and ![](https://www.scirp.org/html/1-5300467\08da99f3-a522-4b2d-9f4b-559d1ba69f32.jpg)
not both zero, such that
almost everywhere in
.
There exists positive integer N such that for
almost everywhere in (x,b). Hence,
and
for
, and also
![](https://www.scirp.org/html/1-5300467\cd67532b-ee9e-4874-97dd-cabd0aa64a34.jpg)
This contradicts the facts that
. The lemma is proved.
Theorem 4 Let
,
be finite non-negative measurable functions on
,
and
with ![](https://www.scirp.org/html/1-5300467\f2172ca0-e590-47c0-932f-3710a8bcaf5c.jpg)
such that
, then the following inequality holds:
(5)
where
![](https://www.scirp.org/html/1-5300467\53f56bf9-edac-4860-9277-243888e60693.jpg)
and
![](https://www.scirp.org/html/1-5300467\89a29a78-ddd3-4a8c-bfe6-84b8f5218f5e.jpg)
Proof
![](https://www.scirp.org/html/1-5300467\caf7c70b-37a8-4bb7-bf70-734085f89d05.jpg)
where C is as stated in the statement of the theorem and this proves the theorem.
The next results are on convex functions as it applies to Hardy-type inequalities.
Lemma. local minimum of a function f is a global minimum if and only if f is strictly convex.
Proof
The necessary part follows from the fact that if a point
is a local optimum of a convex function
. Then
for any
in some neighborhood
of
. For any
,
belongs to
and
sufficiently close to
implies that
is a global optimum. For the sufficient part, we let
be a strictly convex function with convex domain. Suppose
has a local minimum at
and
such that
and assuming
. By strict convexity and for any
, we have,
![](https://www.scirp.org/html/1-5300467\ae3365fa-6a50-413f-8408-857cd255ddf7.jpg)
Since any neighborhood of
contains points of the form
with
, thus the neighborhood of
contains points
for which
. Hence,
does not have a local minimum at
, a contradiction. It must be that
, this shows that
has at most one local minimum.
Lemma. Let
and
. If
is a positive convex function on (a,c), then
(6)
Proof
![](https://www.scirp.org/html/1-5300467\9463f1ce-09d4-4dea-8b16-aa002122fa56.jpg)
Hence the proof.
Lemma. Let
be non-negative for
,
non decreasing and
. then
(7)
Proof
Let
be continuous and convex, If
has a continuous inverse which is neccessarily concave, then by Jensen’s inequality we have
![](https://www.scirp.org/html/1-5300467\775f4c21-7cf0-4d89-ad5a-ca1a74edcaa0.jpg)
Taking
,
, we obtain
![](https://www.scirp.org/html/1-5300467\8237106c-0cb1-4d3c-952a-a480d3a4bee5.jpg)
for
, we have
![](https://www.scirp.org/html/1-5300467\5988e145-b755-40f7-8e35-c2bdf97c69f0.jpg)
which we write as
![](https://www.scirp.org/html/1-5300467\b15d5fc8-4685-4016-a26c-5bc3bd7bab9e.jpg)
This complete the proof.
Theorem 5 If
and
, let f, g be defined on (0,b) such that
, then
(8)
Proof
![](https://www.scirp.org/html/1-5300467\7944e420-8dc3-450e-98fb-e61c262f8841.jpg)
Since
is a convex function, applying Jensen’s inequality to the above gives
![](https://www.scirp.org/html/1-5300467\4ff4e150-101b-4f7c-b658-958544770521.jpg)
The result follows.
Theorem 6 Let g be a continuous and nondecreasing on
,
, with
for
and
. Let
and
be nonnegative and Lebesgue-Stieltjes integrable with respect to
on
. Suppose r is a real number such that
then,
(9)
where
![](https://www.scirp.org/html/1-5300467\a299dc7b-928b-4ab7-82e7-339b9599423d.jpg)
Proof
In the inequality (2.5), we let
![](https://www.scirp.org/html/1-5300467\47852ef7-ad61-43a5-a304-51f0ac5e6457.jpg)
and
![](https://www.scirp.org/html/1-5300467\888d07d7-5ca5-4416-ae63-9312241feb15.jpg)
Then, the left hand side of (2.5) becomes
![](https://www.scirp.org/html/1-5300467\6db54d23-c2a6-4735-b6c7-e3905cc50d80.jpg)
and the right hand side reduces to
![](https://www.scirp.org/html/1-5300467\5ae16e7b-9ce0-4a67-a568-9589e143f6ac.jpg)
Hence, inequality (2.5) becomes
![](https://www.scirp.org/html/1-5300467\d9d18e20-9288-4e39-8df5-2ca9480886b3.jpg)
for
, we have
![](https://www.scirp.org/html/1-5300467\bfa1a7c4-ccc2-46cc-9a2c-748c6efa265f.jpg)
Integrating both sides with respect to
and then raising both sides to power
yields
![](https://www.scirp.org/html/1-5300467\d585363d-88b5-4d81-9872-3e21f07efb34.jpg)
Applying Minkowski integral inequality to the right hand side implies
![](https://www.scirp.org/html/1-5300467\d3916d74-5de2-41f8-b4a6-330060ece890.jpg)
Since ![](https://www.scirp.org/html/1-5300467\bc2ee318-cabe-4e15-97fa-9b6e06ec3265.jpg)
![](https://www.scirp.org/html/1-5300467\2889f8e6-f5ea-47ba-bfc0-a9d6fbb8755b.jpg)
Hence, we have
![](https://www.scirp.org/html/1-5300467\90bbc2e2-f79b-44f8-bac3-590d8bf3cd98.jpg)
Which complete the proof of the Theorem.
3. Conclusion
This work obtained considerable improvement on AdeagboSheikh and Imoru results and applications for measurable and convex functions are also given.