Abstract
The aim of the present paper is to establish some fractional -integral inequalities on the specific time scale,
,
a nonnegative integer
, where
, and
.
Journal of Inequalities and Applications volume 2011, Article number: 787939 (2011)
The aim of the present paper is to establish some fractional -integral inequalities on the specific time scale,
,
a nonnegative integer
, where
, and
.
The study of fractional -calculus in [1] serves as a bridge between the fractional
-calculus in the literature and the fractional
-calculus on a time scale
, where
, and
.
Belarbi and Dahmani [2] gave the following integral inequality, using the Riemann-Liouville fractional integral: if and
are two synchronous functions on
, then
for all ,
.
Moreover, the authors [2] proved a generalized form of (1.1), namely that if and
are two synchronous functions on
, then
for all ,
, and
.
Furthermore, the authors [2] pointed out that if are
positive increasing functions on
, then
for any ,
.
In this paper, we have obtained fractional -integral inequalities, which are quantum versions of inequalities (1.1), (1.2), and (1.3), on the specific time scale
, where
, and
. In general, a time scale is an arbitrary nonempty closed subset of the real numbers [3].
Many authors have studied the fractional integral inequalities and applications. For example, we refer the reader to [4–6].
To the best of our knowledge, this paper is the first one that focuses on fractional -integral inequalities.
Let and define
If there is no confusion concerning , we will denote
by
. For a function
, the nabla
-derivative of
is
for all . The
-integral of
is
The fundamental theorem of calculus applies to the -derivative and
-integral; in particular,
and if is continuous at 0, then
Let ,
denote two time scales. Let
be continuous let
be
-differentiable, strictly increasing, and
. Then for
,
The -factorial function is defined in the following way: if
is a positive integer, then
If is not a positive integer, then
The -derivative of the
-factorial function with respect to
is
and the -derivative of the
-factorial function with respect to
is
The -exponential function is defined as
Define the -Gamma function by
Note that
The fractional -integral is defined as
Note that
More results concerning fractional -calculus can be found in [1, 7–9].
In this section, we will state our main results and give their proofs.
Theorem 3.1.
Let and
be two synchronous functions on
. Then for all
,
, we have
Proof.
Since and
are synchronous functions on
, we get
for all ,
. By (3.2), we write
Multiplying both side of (3.3) by , we have
Integrating both sides of (3.4) with respect to on
, we obtain
So,
Hence, we have
Multiplying both side of (3.7) by , we obtain
Integrating both side of (3.8) with respect to on
, we get
Obviously,
and the proof is complete.
The following result may be seen as a generalization of Theorem 3.1.
Theorem 3.2.
Let and
be as in Theorem 3.1. Then for all
,
,
we have
Proof.
By making similar calculations as in Theorem 3.1 we have
Integrating both side of (3.12) with respect to on
, we obtain
Thus, (3.11) holds for all ,
,
, so the proof is complete.
Remark 3.3.
The inequalities (3.1) and (3.11) are reversed if the functions are asynchronous on (i.e.,
, for any
).
Theorem 3.4.
Let be
positive increasing functions on
. Then for any
,
we have
Proof.
We prove this theorem by induction.
Clearly, for , we have
for all ,
.
For , applying (3.1), we obtain
for all ,
.
Suppose that
Since are positive increasing functions, then
is an increasing function. Hence, we can apply Theorem 3.1 to the functions
,
. We obtain
Taking into account the hypothesis (3.17), we obtain
and this ends the proof.
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The authors thank referees for suggestions which have improved the final version of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Öğünmez, H., Özkan, U. Fractional Quantum Integral Inequalities. J Inequal Appl 2011, 787939 (2011). https://doi.org/10.1155/2011/787939
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DOI: https://doi.org/10.1155/2011/787939