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Fractional Quantum Integral Inequalities
Journal of Inequalities and Applications volume 2011, Article number: 787939 (2011)
Abstract
The aim of the present paper is to establish some fractional -integral inequalities on the specific time scale,
,
a nonnegative integer
, where
, and
.
1. Introduction
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ1_HTML.gif)
for all ,
.
Moreover, the authors [2] proved a generalized form of (1.1), namely that if and
are two synchronous functions on
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ2_HTML.gif)
for all ,
, and
.
Furthermore, the authors [2] pointed out that if are
positive increasing functions on
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ3_HTML.gif)
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.
2. Description of Fractional
-Calculus
Let and define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ4_HTML.gif)
If there is no confusion concerning , we will denote
by
. For a function
, the nabla
-derivative of
is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ5_HTML.gif)
for all . The
-integral of
is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ6_HTML.gif)
The fundamental theorem of calculus applies to the -derivative and
-integral; in particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ7_HTML.gif)
and if is continuous at 0, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ8_HTML.gif)
Let ,
denote two time scales. Let
be continuous let
be
-differentiable, strictly increasing, and
. Then for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ9_HTML.gif)
The -factorial function is defined in the following way: if
is a positive integer, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ10_HTML.gif)
If is not a positive integer, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ11_HTML.gif)
The -derivative of the
-factorial function with respect to
is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ12_HTML.gif)
and the -derivative of the
-factorial function with respect to
is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ13_HTML.gif)
The -exponential function is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ14_HTML.gif)
Define the -Gamma function by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ15_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ16_HTML.gif)
The fractional -integral is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ17_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ18_HTML.gif)
More results concerning fractional -calculus can be found in [1, 7–9].
3. Main Results
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ19_HTML.gif)
Proof.
Since and
are synchronous functions on
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ20_HTML.gif)
for all ,
. By (3.2), we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ21_HTML.gif)
Multiplying both side of (3.3) by , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ22_HTML.gif)
Integrating both sides of (3.4) with respect to on
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ23_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ24_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ25_HTML.gif)
Multiplying both side of (3.7) by , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ26_HTML.gif)
Integrating both side of (3.8) with respect to on
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ27_HTML.gif)
Obviously,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ29_HTML.gif)
Proof.
By making similar calculations as in Theorem 3.1 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ30_HTML.gif)
Integrating both side of (3.12) with respect to on
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ32_HTML.gif)
Proof.
We prove this theorem by induction.
Clearly, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ33_HTML.gif)
for all ,
.
For , applying (3.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ34_HTML.gif)
for all ,
.
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ35_HTML.gif)
Since are positive increasing functions, then
is an increasing function. Hence, we can apply Theorem 3.1 to the functions
,
. We obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ36_HTML.gif)
Taking into account the hypothesis (3.17), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F787939/MediaObjects/13660_2010_Article_2360_Equ37_HTML.gif)
and this ends the proof.
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The authors thank referees for suggestions which have improved the final version of this paper.
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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