On Some Integral Inequalities of Hardy-Type Operators

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.


Introduction
The classical hardy integral inequality reads: be a non-negative p-integrable function defined on  , and .Then, is integrable over the interval x for each x and the following inequality: holds, where 1 is the best possible constant (see [1]).
This inequality can be found in many standard books (see [2][3][4][5][6][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].
 and   p fg L X  be finite, non-negative measurable functions on   0,  , 0 t a b      and 1 1 1 1 p q r    with 1 p q     such that a x .Then, the following inequality holds: where,


. n [10] also proved the following integral inequality of Hardy-type mainly by Jensen's Inequality: Theorem 3 Let g be continuous and where, , , , , 0.
Other recent developments of the Hardy-type inequalities can be seen in the papers [11][12][13][14][15][16].In this article, we point out some other Hardy-type inequalities which will complement the above results (2) and (3).

Main Results
The following lemma is of particular interest (see also [8]).
 by Holder's inequality, we have, We need to show that there exists not both zero, such that   There exists positive integer N such that for , and also This contradicts the facts th .
The lemma is prov   , then the following inequality holds: where 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 x is a local optimum of a convex function .Then for any z in some neighborhood of U x .For any , y   belongs to U and 1   sufficiently close to 1 i plies that m x is a global optimum.For the sufficient part, we let f be a trictly convex function with convex domain.Suppose Since any neighborhood of contains points of the form does not have a local minimum at , a contradiction.It must be that , this shows that has at most one local minimum.
Proof Let be continuous and convex, If has a continuous inverse which is neccessarily concave, then by Jensen's inequality we have The result follows.
f x be nonne-gative and Lebesgue-Stieltjes integrable with respect to where Proof and In the inequality (2.5), we let Then, the left hand side of (2.5) becomes Hence, inequality (2.5) becomes Integrating both sides with respect to   g x and then raising both sides to power Applying Minkowski integral inequality to the right hand side implies  C a b p q r g x f x g x Hence, we have

. Conclusion
This work obtained considerable improvement on Adeagbo-Sheikh and Imoru results and applications for measurable also given.
Which complete the proof of the Theorem.