Open Access

Fractional Quantum Integral Inequalities

Journal of Inequalities and Applications20112011:787939

Received: 10 November 2010

Accepted: 16 February 2011

Published: 10 March 2011


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

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 [46].

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
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, 79].

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


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
Hence, we have
Multiplying both side of (3.7) by , we obtain
Integrating both side of (3.8) with respect to on , we get

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


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


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.



The authors thank referees for suggestions which have improved the final version of this paper.

Authors’ Affiliations

Department of Mathematics, Faculty of Science and Arts, Kocatepe University


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© H. Öğünmez and U. M. Özkan. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.