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.