International Journal of Analysis

Volume 2013, Article ID 127061, 5 pages

http://dx.doi.org/10.1155/2013/127061

## On Right Caputo Fractional Ostrowski Inequalities Involving Three Functions

Department of Mathematics, Dr. B.A.M. University, Aurangabad, Maharashtra 431004, India

Received 28 August 2012; Accepted 27 December 2012

Academic Editor: Abdallah El Hamidi

Copyright © 2013 Deepak B. Pachpatte. 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.

#### Abstract

We establish Ostrowski inequalities involving three functions in right Caputo fractional derivative in spaces.

#### 1. Introduction

In 1938, Ostrowski proved the following useful inequality.

Let be continuous on and differentiable on whose derivative is bounded on , that is, . Then for any . The constant is best possible.

In [1, 2] Pachpatte has proved Ostrowski inequality in three independent variables.

In past few years many authors have obtained various generalisation and variant of the above type of inequality and other on fractional as well as time scale calculus see [3–6].

Here we give some basic definition from fractional calculus used in [7–9].

*Definition 1. *Let , . The right and left Riemann-Liouville integrals and of order with are defined by
respectively, where and .

*Definition 2 (see [10, page 2]). *Let ( be in ), , , ( the ceiling of the number). We define the right Caputo fractional derivative of order by

If , then

If , we define .

*Definition 3 (see [9, page 74]). *Let , with , and then the fractional integral is defined by

We give here the theorems proved in [10].

Theorem 4. *Let , , . Assume that , and . Then
*

Theorem 5. *Let , , . Assume that , and . Then
*

Theorem 6. *Let ; , , , . Assume that , and . Then
*

#### 2. Main Results

Our main results are given in the following theorems.

Theorem 7. *Let , , . Assume that , and . Then
*

*Proof. *Let we have
Multiplying (10), (11), and (12) by , , and , respectively, and adding them, we have
Integrating both sides of (13) with respect to and rewriting above equation we have
From (14) and using the properties of modulus we have
It is easy to observe that
The proof of the theorem is complete.

*Remark 8. *If we take , and hence , in Theorem 7, then we get Theorem 4.

Theorem 9. *Let , , . Assume that , , and . Then
*

*Proof. *From (15) we have

This proves the theorem.

*Remark 10. *If we take , , and hence , in Theorem 9, then we get Theorem 5.

Theorem 11. *Let ; , , , . Assume that , , and . Then
*

*Proof. *From (15) we have

Applying Holder’s inequality to (21), we get

We have

Substituting (22) into (21), we get the required inequality.

*Remark 12. *If we take and hence , in Theorem 11, then we get Theorem 6.

#### References

- B. G. Pachpatte, “On an inequality of Ostrowski type in three independent variables,”
*Journal of Mathematical Analysis and Applications*, vol. 249, no. 2, pp. 583–591, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. G. Pachpatte, “New Ostrowski and Grüss type inequalities,”
*Analele Stintifice Ale Universitath Al.I Cuza Iasi, Tomul LI. s. I Mathematica*, vol. 51, no. 2, pp. 377–386, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. A. Anastassiou,
*Intelligent Mathematics: Computational Analysis*, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. A. Anastassiou, “On right fractional calculus,”
*Chaos, Solitons and Fractals*, vol. 42, no. 1, pp. 365–376, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. A. Bohner, M. Bohner, and T. Matthews, “Time scales ostrowski and gruss type inequalities involving three functions,”
*Dynamics and Systems Theory*, vol. 12, no. 2, pp. 119–135, 2012. View at Google Scholar - M. Bohner and T. Matthews, “Ostrowski inequalities on time scales,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 1, article 6, 8 pages, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. A. Anastassiou,
*Fractional Differentiation Inequalities*, Springer, Dordrecht, The Netherlands, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet - G. A. Anastassiou,
*Advances in Fractional Inequalities*, Springer, New York, NY, USA, 2011.