Research Article | Open Access
Deepak B. Pachpatte, "On Right Caputo Fractional Ostrowski Inequalities Involving Three Functions", International Journal of Analysis, vol. 2013, Article ID 127061, 5 pages, 2013. https://doi.org/10.1155/2013/127061
On Right Caputo Fractional Ostrowski Inequalities Involving Three Functions
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
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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.