International Journal of AnalysisVolume 2013 (2013), Article ID 127061, 5 pageshttp://dx.doi.org/10.1155/2013/127061

Research Article

On Right Caputo Fractional Ostrowski Inequalities Involving Three Functions

Deepak B. Pachpatte

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.

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.

G. A. Anastassiou, Intelligent Mathematics: Computational Analysis, Springer, Berlin, Germany, 2011.

G. A. Anastassiou, “On right fractional calculus,” Chaos, Solitons and Fractals, vol. 42, no. 1, pp. 365–376, 2009.

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.

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.

G. A. Anastassiou, Fractional Differentiation Inequalities, Springer, Dordrecht, The Netherlands, 2010.

I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

G. A. Anastassiou, Advances in Fractional Inequalities, Springer, New York, NY, USA, 2011.