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
with such that. Then, the following inequality holds:
(2)
where,
and
[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
Suppose is a real number such that then
(3)
where,
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
then,
by Holder’s inequality, we have,
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
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
This contradicts the facts that. The lemma is proved.
Theorem 4 Let, be finite non-negative measurable functions on,
and with
such that, then the following inequality holds:
(5)
where
and
Proof
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,
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
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
Taking, , we obtain
for, we have
which we write as
This complete the proof.
Theorem 5 If and, let f, g be defined on (0,b) such that, then
(8)
Proof
Since is a convex function, applying Jensen’s inequality to the above gives
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
Proof
In the inequality (2.5), we let
and
Then, the left hand side of (2.5) becomes
and the right hand side reduces to
Hence, inequality (2.5) becomes
for, we have
Integrating both sides with respect to and then raising both sides to power yields
Applying Minkowski integral inequality to the right hand side implies
Since
Hence, we have
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.