On new inequalities via Riemann-Liouville fractional integration

In this paper, we extend the Montogomery identities for the Riemann-Liouville fractional integrals. We also use this Montogomery identities to establish some new integral inequalities for convex functions.


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 f : a, b → R is a differentiable function with bounded derivative, then for every x ∈ a, b . Moreover, the constant 1/4 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 f : a, b → R is differentiable on a, b with the first derivative f integrable on a, b , then Montgomery identity holds 1 x, t f t dt, 1.2 where P 1 x, t is the Peano kernel defined by P 1 x, t : 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 n-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.

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 α ≥ 0 with a ≥ 0 is defined as

2.1
Recently, many authors have studied a number of inequalities by using the Riemann-Liouville fractional integrals, see 13-16 and the references cited therein.

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.
Abstract and Applied Analysis 3 where P 2 x, t is the fractional Peano kernel defined by

3.3
Integrating by parts, we can state 3.4 and similarly, Adding 3.4 and 3.5 , we get

4 Abstract and Applied Analysis
If we add and subtract the integral to the right-hand side of the equation above, then we have and so This completes the proof. Now, we extend Lemma 3.1 as follows.
Theorem 3.2. Let f : I ⊂ R → R be a differentiable function on I • with f ∈ L 1 a, b , then the following identity holds:

3.10
Abstract and Applied Analysis 5 where P 3 x, t is the fractional Peano kernel defined by Proof. By similar way in proof of Lemma 3.1, we have

3.12
Integrating by parts, we can state

3.13
and similarly, 3.14 Thus, by using J 1 and J 2 in 3.12 , we get 3.10 which completes the proof.

Remark 3.3.
We note that in the special cases, if we take λ 0 in Theorem 3.2, then we get 3.1 with the kernel P 2 x, t .
If |f x | ≤ M for every x ∈ a, b and α ≥ 1, then the following inequality holds:

3.15
Proof. From Theorem 3.2, we get

3.16
By simple computation, we obtain

3.17
Abstract and Applied Analysis 7 and similarly

3.18
By using J 3 and J 4 in 3.16 , we obtain 3.15 . Then for any x ∈ a, b , the following inequality holds:

3.19
Proof. Similarly to the proof of Lemma 3.1, we have 3.20 Since f is convex, then for any x ∈ a, b we have the following inequalities: If we multiply 3.21 by b − t α−1 t − a ≥ 0, t ∈ a, x , α ≥ 1 and integrate on a, x , we 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 α 1, 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 x a b /2 in Corollary 3.7, we get Theorem 3.9. Let f : a, b → R be a differentiable convex function on a, b and f ∈ L 1 a, b . Then for any x ∈ a, b , the following inequality holds:

3.27
Proof. Assume that f a and f − b are finite. Since f is convex on a, b , then we have the following inequalities: f t ≥ f a for a.e. t ∈ a, x , 3.28

3.29
Abstract and Applied Analysis 9 If we multiply 3.28 by b − t α−1 t − a ≥ 0, t ∈ a, x , α ≥ 1 and integrate on a, x , we have

3.30
and if we multiply 3.29 by b − t α ≥ 0, t ∈ x, b , α ≥ 1 and integrate on x, b , 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 .
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 x a b /2 in Corollary 3.10, we get 3.33