ABSTRACT

In recent time, hardy integral inequalities have received attentions of many researchers. The aim of this paper is to obtain new integral inequalities of hardy-type which complement some recent results.

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.

REFERENCES