1. Introduction
Some inequality were in their reigns such as Wirtinger’s, Holder’s, Cauchy’s, Minkwoski’s, Hardy’s and Opial’s inequalities. Opial [1] established an inequality involving integral of a function and it’s derivatives as follows:
Theorem 1.1. Let
be such that
, and
. Then, the following inequality holds:
(1.2)
in (1.2), the constant
is the best possible.
This inequality and it’s generalisations have various applications in the theories of differential and difference equations.
In [2] , Olech simplified the proof and noted that the positive requirement of
in (1.2) is unnecessary and that inequality (1.2) holds absolutely continuous in
. He however, demonstrated the following result:
Theorem 1.3. Let
be absolutely continuous in
such that
and
in
with
. Then the following inequality holds.
(1.4)
In [3] , Hardy obtained the following result:
Theorem 1.5. If
, and
, then
(1.6)
unless
. The constant
is the best possible.
By using Holder’s inequality, Beesack [4] gave a refinement of Opial-type inequalities by establishing the conditions under which the weighted Opial-type of (1.2) holds
(1.7)
for all l with
, where
and
. The case of negative values of
and s was also considered.
One of the special cases of (1.7) when
and
is:
(1.8)
and the weighted Hardy inequality is
(1.9)
If
, then (1.7) reduces to weighted Hardy inequality (1.9).
In [5] , Imoru and Adeagbo-Sheikh established the following result:
Theorem 1.10. Let
be continuous and non-decreasing on
,
, with
for
,
and
be non-negative Lebesgue-Stieltjes integrable with respect to
on
. If
is a real number such that
, then
(1.11)
where
(1.12)
Oguntuase [6] presented an integral inequality by using Hölder’s inequality to obtain an integral inequality that has Opial’s and Hardy’s inequalities as special cases. However, he observed that constant (1.12) at the right hand side was wrongly written and obtained a better constant as stated below:
Theorem 1.13. Let
be continuous and non-decreasing on
,
, with
for
. Let
and
be non-negative and Lebesgue-Stieltjes integrable with respect to
on
.
Suppose
is a real number such that
then
(1.14)
where
(1.15)
In [7] , Adeagbo-Sheikh and Fabelurin follow the trend by using Jensen’s and Minkowski’s integral inequalities to establish a result that generalized (1.14) as follows:
Theorem 1.16. Let g be a continuous function and non-decreasing on
,
, with
for
. Suppose that
,
,
and
is non-negative, non-decreasing and Lebesgue-Stieltjes integrable with respect to
on
. Then
(1.17)
where
(1.18)
Throughout what follows, unless otherwise stated, we shall assume
to be continuous, non-negative and non-decreasing. Furthermore, f is assumed to be a non-negative integrable function with respect to
.
The inequality on time scale was introduced by Hilger [8] in order to unify discrete and continuous analysis. Hence, a time scale is an arbitrary non empty closed subset of real numbers. Thus, the real numbers, integers, natural numbers and non negative integers are examples of time scales. However, little work has been done on Opial’s inequality on time scales.
A time scale
is an arbitrary nonempty closed subset of the set
of all real numbers. Let
be the topology inherits from standard topology on
. For
, if
, we define the forward jump operator
by
while if
, the backward jump operator
is defined by
If
, we say t is right scattered, while if
, we say t is left scattered.
If
, we say t is right dense, while if
, we say t is left dense.
Throughout this paper, we let:
a)
;
b)
is a time scale and;
c) an interval means the intersection of a real interval with the given time scale.
If
, then
is defined by
A mapping
is called rd-continuous if it satisfies:
a)
is continuous at each right-dense point or maximal element of
;
b) the left-sided
exists at each left-dense point t of
;
c) an interval means the intersection of a real interval with the given time scale;
d)
e)
f) If
, then
is defined by
Let
and fix
, then we define
to be the number (provided it exists) with the property that for any given
, there is a neighborhood U of t such that
for all
. We call
the delta derivative of
at t. It can be shown that if
is continuous at
and t is right scattered, then
A function
is an anti-derivative of
if
for all
. In this case, we define the integral of f by
for
.
The objective of this paper is to obtain a new integral inequality which is an extension of Theorem 1.1, Theorem 1.3, Theorem 1.5 and Theorem 1.13.
Indeed, we shall show that Theorem 1.1 in its modified form leads to some extensions, variants and a new generalization of a class of inequalities which are related to Hardy’s and Opial’s integral inequalities. Moreover, we shall examine the case when
.
In fact, some extensions of Hardy’s inequality due to Imoru [9] are shown to be immediate consequences of the modified form of Theorem 1.3.
The methodology adopted shall be the following:
Let
and
. Suppose
and
,
,
,
,
be non-decreasing and
. Suppose
has a continuous inverse
which is necessarily concave, then
which can be written in the form:
(1.19)
and for
. The later inequality is the Jensen’s inequality for convex functions and putting convex function
in (1.19) yields:
and
However, the validity of the left hand side of (1.19) solely depends on the right hand side.
2. Statement of the Main Result
In this section, we shall show the main result in the following theorems:
Theorem 2.1. Let
and
. Suppose
and
. Let
and
be absolutely continuous and non-decreasing on
and
with
for
and
,
,
and
.
and
with
and
. If
then we have,
(2.2)
whenever
or
. The inequality is reversed if
. The inequality is strict unless either
or
.
The constant factor
is the best possible when the term
is valid throughout.
2.1. Remark
Let
and
. By applying Cauchy’s inequality with exponent conjugates
and
, we obtain the following results.
(2.3)
2.2. Remark
Let
and observe that when
, taking
in (2.3), then (2.3) reduces to Oguntuase’s result.
The modified Minkowski integral inequalities (2.5) and (2.6) are stated in Opic and Kufner [10] on page 21 and 22 as follows:
Lemma 2.4. Let
be a non negative measurable function on
and let
Then,
(2.5)
If
where
and
then (2.5) reduces to
(2.6)
The proof of our main results will depend essentially on the following lemmas:
Lemma 2.7. Let
and
. Suppose
and
with
which satisfied the following inequality:
(2.8)
and the other side of (1.19) implies the following:
Lemma 2.9. Let
and
. Suppose
and
with
which also satisfied the inequality:
(2.10)
2.3. Remark
The Proof of Lemma 2.7 is immediate from the analysis of Lemma 2.9 and hence omitted.
Proof of Theorem
The proof follows from Lemma 2.7 and Lemma 2.9 by combining (2.8) and (2.10) with further simplification as:
(2.11)
integrate both sides of (2.11) with respect to
from 0 to
(2.12)
combining (2.6), (2.8) and (2.10) we have the following inequality:
(2.13)
which implies
(2.14)
2.4. Remark
Let
. The above inequality becomes
3. Discussion of the Results
We refined some existing results on an integral inequality of Hardy-type on Time-scale and obtained the best possible constant by employing modified Minkowski integral inequalities with some standard lemmas. The results extended and generalized some earlier results in literature.