Skip to main content


You are viewing the new article page. Let us know what you think. Return to old version

Research Article | Open | Published:

Fractional Quantum Integral Inequalities


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.


  1. 1.

    Atıcı FM, Eloe PW: Fractional -calculus on a time scale. Journal of Nonlinear Mathematical Physics 2007,14(3):341–352.

  2. 2.

    Belarbi S, Dahmani Z: On some new fractional integral inequalities. Journal of Inequalities in Pure and Applied Mathematics 2009,10(3, article 86):-5.

  3. 3.

    Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2001:x+358.

  4. 4.

    Denton Z, Vatsala AS: Fractional integral inequalities and applications. Computers & Mathematics with Applications 2010,59(3):1087–1094.

  5. 5.

    Anastassiou GA: Multivariate fractional Ostrowski type inequalities. Computers & Mathematics with Applications 2007,54(3):434–447. 10.1016/j.camwa.2007.01.024

  6. 6.

    Anastassiou GA: Opial type inequalities involving fractional derivatives of two functions and applications. Computers & Mathematics with Applications 2004,48(10–11):1701–1731. 10.1016/j.camwa.2003.08.013

  7. 7.

    Agarwal RP: Certain fractional -integrals and -derivatives. Proceedings of the Cambridge Philosophical Society 1969, 66: 365–370. 10.1017/S0305004100045060

  8. 8.

    Al-Salam WA: Some fractional -integrals and -derivatives. Proceedings of the Edinburgh Mathematical Society. Series II 1966, 15: 135–140. 10.1017/S0013091500011469

  9. 9.

    Rajković PM, Marinković SD, Stanković MS: A generalization of the concept of -fractional integrals. Acta Mathematica Sinica (English Series) 2009,25(10):1635–1646. 10.1007/s10114-009-8253-x

Download references


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

Author information

Correspondence to UmutMutlu Özkan.

Rights and permissions

Reprints and Permissions

About this article


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