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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

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.

## 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, 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

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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## Acknowledgment

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

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### Keywords

- Positive Integer
- Real Number
- Exponential Function
- Specific Time
- Similar Calculation