- Research Article
- Open Access

# Fractional Quantum Integral Inequalities

- Hasan Öğünmez
^{1}and - UmutMutlu Özkan
^{1}Email author

**2011**:787939

https://doi.org/10.1155/2011/787939

© H. Öğünmez and U. M. Özkan. 2011

**Received:**10 November 2010**Accepted:**16 February 2011**Published:**10 March 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 .

## Keywords

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

## 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 .

for all , .

for all , , and .

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

## 3. Main Results

In this section, we will state our main results and give their proofs.

Theorem 3.1.

Proof.

and the proof is complete.

The following result may be seen as a generalization of Theorem 3.1.

Theorem 3.2.

Proof.

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.

Proof.

We prove this theorem by induction.

for all , .

for all , .

and this ends the proof.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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## Copyright

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.