Abstract and Applied Analysis

Volume 2012 (2012), Article ID 428983, 10 pages

doi:10.1155/2012/428983

Research Article

On New Inequalities via Riemann-Liouville Fractional Integration

Mehmet Zeki Sarikaya¹ and Hasan Ogunmez²

¹Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

²Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon, Turkey

Received 9 August 2012; Accepted 6 October 2012

Academic Editor: Ciprian A. Tudor

Copyright © 2012 Mehmet Zeki Sarikaya and Hasan Ogunmez. 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 extend the Montgomery identities for the Riemann-Liouville fractional integrals. We also use these Montgomery identities to establish some new integral inequalities. Finally, we develop some integral inequalities for the fractional integral using differentiable convex functions.

1. Introduction

The inequality of Ostrowski [1] gives us an estimate for the deviation of the values of a smooth function from its mean value. More precisely, if is a differentiable function with bounded derivative, then

for every . Moreover, the constant is the best possible.

For some generalizations of this classic fact see ([2], (pages 468–484)) by Mitrinović et al. A simple proof of this fact can be done by using the following identity [2].

If is differentiable on with the first derivative integrable on , then Montgomery identity holds

where is the Peano kernel defined by

Recently, several generalizations of the Ostrowski integral inequality are considered by many authors; for instance, covering the following concepts: functions of bounded variation, Lipschitzian, monotonic, absolutely continuous, and -times differentiable mappings with error estimates with some special means together with some numerical quadrature rules. For recent results and generalizations concerning Ostrowski’s inequality, we refer the reader to the recent papers [3–10].

In this paper, we extend the Montgomery identities for the Riemann-Liouville fractional integrals. We also use these Montgomery identities to establish some new integral inequalities of Ostrowski's type. Finally, we develop some integral inequalities for the fractional integral using differentiable convex functions. Later, we develop some integral inequalities for the fractional integral using differentiable convex functions. From our results, the weighted and the classical Ostrowski's inequalities can be deduced as some special cases.

2. Fractional Calculus

Firstly, we give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used further in this paper. For more details, one can consult [11, 12].

Definition 2.1. The Riemann-Liouville fractional integral operator of order with is defined as

Recently, many authors have studied a number of inequalities by using the Riemann-Liouville fractional integrals, see [13–16] and the references cited therein.

3. Main Results

In order to prove some of our results, by using a different method of proof, we give the following identities, which are proved in [13]. Later, we will generalize the Montgomery identities in the next theorem.

Lemma 3.1. Let be a differentiable function on with and , then

where is the fractional Peano kernel defined by

Proof. By definition of , we have

Integrating by parts, we can state

and similarly,

Adding (3.4) and (3.5), we get

If we add and subtract the integral to the right-hand side of the equation above, then we have

Multiplying both sides by , we obtain

and so

This completes the proof.

Now, we extend Lemma 3.1 as follows.

Theorem 3.2. Let be a differentiable function on with , then the following identity holds:

where is the fractional Peano kernel defined by

for .

Proof. By similar way in proof of Lemma 3.1, we have

Integrating by parts, we can state

and similarly,

Thus, by using and in (3.12), we get (3.10) which completes the proof.

Remark 3.3. We note that in the special cases, if we take in Theorem 3.2, then we get (3.1) with the kernel .

Theorem 3.4. Let be a differentiable on such that , where . If for every and , then the following inequality holds:

Proof. From Theorem 3.2, we get

By simple computation, we obtain

and similarly

By using and in (3.16), we obtain (3.15).

Remark 3.5. If we take in Theorem 3.4, then it reduces Theorem 4.1 proved by Anastassiou et al. [13]. So, our results are generalizations of the corresponding results of Anastassiou et al. [13].

Theorem 3.6. Let be a differentiable convex function on and . Then for any , the following inequality holds:

Proof. Similarly to the proof of Lemma 3.1, we have

Since is convex, then for any we have the following inequalities:

If we multiply (3.21) by , and integrate on , we get

and if we multiply (3.22) by and integrate on , we also get

Finally, if we subtract (3.24) from (3.23) and use the representation (3.20) we deduce the desired inequality (3.19).

Corollary 3.7. Under the assumptions Theorem 3.6 with , one has

The proof of Corollary 3.7 is proved by Dragomir in [6]. Hence, our results in Theorem 3.6 are generalizations of the corresponding results of Dragomir [6].

Remark 3.8. If we take in Corollary 3.7, we get

Theorem 3.9. Let be a differentiable convex function on and . Then for any , the following inequality holds:

Proof. Assume that and are finite. Since is convex on , then we have the following inequalities:

If we multiply (3.28) by and integrate on , we have

and if we multiply (3.29) by , and integrate on , we also have

Finally, if we subtract (3.30) from (3.31) and use the representtation (3.20) we deduce the desired inequality (3.27).

Corollary 3.10. Under the assumptions Theorem 3.9 with , one

The proof of Corollary 3.10 is proved by Dragomir in [6]. So, our results in Theorem 3.9 are generalizations of the corresponding results of Dragomir [6].

Remark 3.11. If we take in Corollary 3.10, we get

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