- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
International Journal of Analysis
Volume 2013 (2013), Article ID 127061, 5 pages
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.
We establish Ostrowski inequalities involving three functions in right Caputo fractional derivative in spaces.
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.
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 .
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.
Theorem 9. Let , , . Assume that , , and . Then
Proof. From (15) we have
This proves the theorem.
Theorem 11. Let ; , , , . Assume that , , and . Then
- 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.