International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 869434, 6 pages

http://dx.doi.org/10.1155/2014/869434

Research Article

Some New Riemann-Liouville Fractional Integral Inequalities

Jessada Tariboon,¹ Sotiris K. Ntouyas,² and Weerawat Sudsutad¹

¹Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok, Thailand

²Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 12 December 2013; Accepted 23 January 2014; Published 10 March 2014

Academic Editor: Ram N. Mohapatra

Copyright © 2014 Jessada Tariboon et al. 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.

In this paper, some new fractional integral inequalities are established.

1. Introduction

In [1] (see also [2]), the Grüss inequality is defined as the integral inequality that establishes a connection between the integral of the product of two functions and the product of the integrals. The inequality is as follows.

If and are two continuous functions on satisfying and for all , , then

The literature on Grüss type inequalities is now vast, and many extensions of the classical inequality were intensively studied by many authors. In the past several years, by using the Riemann-Liouville fractional integrals, the fractional integral inequalities and applications have been addressed extensively by several researchers. For example, we refer the reader to [3–9] and the references cited therein. Dahmani et al. [10] gave the following fractional integral inequalities by using the Riemann-Liouville fractional integrals. Let and be two integrable functions on satisfying the following conditions:

For all , , , then

In this paper, we use the Riemann-Liouville fractional integrals to establish some new fractional integral inequalities of Grüss type. We replace the constants appeared as bounds of the functions and , by four integrable functions. From our results, the above inequalities of [10] and the classical Grüss inequalities can be deduced as some special cases.

In Section 2 we briefly review the necessary definitions. Our results are given in Section 3. The proof technique is close to that presented in [10]. But the obtained results are new and also can be applied to unbounded functions as shown in examples.

2. Preliminaries

Definition 1. The Riemann-Liouville fractional integral of order of a function is defined by

where is the gamma function.

For the convenience of establishing our results, we give the semigroup property:

which implies the commutative property

From Definition 1, if , then we have

3. Main Results

Theorem 2. Let be an integrable function on . Assume that

there exist two integrable functions , on such that

Then, for , , one has

Proof. From , for all , , we have

Therefore

Multiplying both sides of (11) by , , we get

Integrating both sides of (12) with respect to on , we obtain

which yields

Multiplying both sides of (14) by , , we have

Integrating both sides of (15) with respect to on , we get

Hence, we deduce inequality (9) as requested. This completes the proof.

As a special case of Theorem 2, we obtain the following result.

Corollary 3. Let be an integrable function on satisfying , for all and . Then, for and , one has

Example 4. Let be a function satisfying for . Then, for and , we have

Theorem 5. Let and be two integrable functions on . Suppose that holds and moreover one assumes that

  there exist and integrable functions on such that

Then, for , , the following inequalities hold:

Proof. To prove , from and , we have for that

Therefore

Multiplying both sides of (22) by , , we get

Integrating both sides of (23) with respect to on , we obtain

Then we have

Multiplying both sides of (25) by , , we have

Integrating both sides of (26) with respect to on , we get the desired inequality .

To prove , we use the following inequalities:

As a special case of Theorem 5, we have the following corollary.

Corollary 6. Let and be two integrable functions on . Assume that

  there exist real constants such that

Then, for , , we have

Lemma 7. Let be an integrable function on and let , be two integrable functions on . Assume that the condition holds. Then, for , , we have

Proof. For any and , we have

Multiplying (31) by , , and integrating the resulting identity with respect to , from to , we get

Multiplying (32) by , , and integrating the resulting identity with respect to , from to , we have

which implies (30).

If and , , for all , then inequality (30) reduces to the following corollary [10, Lemma 3.2].

Corollary 8. Let be an integrable function on satisfying , for all . Then, for all , , one has

Theorem 9. Let and be two integrable functions on and let , , , and be four integrable functions on satisfying the conditions and on . Then, for all , , one has

where is defined by

Proof. Let and be two integrable functions defined on satisfying and . Define

Multiplying both sides of (37) by , and integrating the resulting identity with respect to and , from to , we can state that

Applying the Cauchy-Schwarz inequality to (38), we have

Since and , for , we have

Thus, from Lemma 7, we get

From (39), (41), and (42), we obtain (35).

Remark 10. If and , , then inequality (35) reduces to

See [10, Theorem 3.1].

Example 11. Let and be two functions satisfying and for . Then, for and , we have

where

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was funded by King Mongkut’s University of Technology North Bangkok, Thailand. Project code: KMUTNB-GRAD-56-02. Sotiris K. Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

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