Montgomery Identity and Ostrowski Type Inequalities for Riemann-Liouville Fractional Integral

Received 31 May 2014; Accepted 27 August 2014; Published 10 September 2014 Academic Editor: Ralf Meyer Copyright © 2014 Andrea Aglić Aljinović. 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 present Montgomery identity for Riemann-Liouville fractional integral as well as for fractional integral of a function f with respect to another function g. We further use them to obtain Ostrowski type inequalities involving functions whose first derivatives belong to Lp spaces. These inequalities are generally sharp in case p > 1 and the best possible in case p = 1. Application for Hadamard fractional integrals is given.


Introduction
The following Ostrowski inequality is well known [1]:        ()      < +∞. ( Ostrowski proved this inequality in 1938, and since then it has been generalized in a number of ways.Over the last few decades, some new inequalities of this type have been intensively considered together with their applications in numerical analysis, probability, information theory, and so forth.This inequality can easily be proved by using the following Montgomery identity (see, for instance, [2]): where (, ) is the Peano kernel, defined by The Riemann-Liouville fractional integral of order  > 0 is defined by where Γ() is a gamma function When  = 0 (5) is the Riemann definition of fractional integral.In case  = 1, fractional integral reduces to classical integral.
In [3], the following Montgomery identity for fractional integrals is obtained.
and  ∈ [, ].Then, the following identity holds: where In [3], the authors used this identity to obtain the following Ostrowski type inequality for fractional integrals.
These results were further generalized in [4], while in [5] generalizations are obtained for fractional integral of a function  with respect to another function  (defined in Section 3).
In the present paper, we give another, simpler new generalization of Montgomery identity for Riemann-Liouville fractional integral of order , which holds for a larger set of ; that is,  > 0. We also obtain Montgomery identity for fractional integral of a function  with respect to another function .We further use these identities to obtain generalizations of Ostrowski inequality for fractional integrals of a function  with respect to another function  for functions whose first derivatives   belong to   spaces.These inequalities are generally sharp in case  > 1 and the best possible in case  = 1.As a special case, application for Hadamard fractional integrals is given.

Montgomery Identity for Fractional Integrals
In this section we give another, simpler new generalization of Montgomery identity for fractional integrals by using the weighted Montgomery identity. where and     () is the Riemann-Liouville integral operator of order  > 0 defined by (5).

Montgomery Identity for Fractional Integral of a Function 𝑓 with respect to Another Function 𝑔
Let  > 0,  an increasing function on (, ], and   a continuous function on (, ).The fractional integral of a function  with respect to another function  is given by In case () = ,  ∈ [, ],    ; () reduces to the Riemann-Liouville integral operator     ().
Theorem 5. Suppose that all the assumptions of Theorem 3 hold.Additionally, assume that  is an increasing function on (, ] and   a continuous function on (, ).Then, the following identity holds: where is the fractional integral of order  > 0 of a function  with respect to another function .
Proof.Apply (10) with normalized weighted function and hence We obtain

New Ostrowski Type Inequalities
In this section, using the Montgomery identity for fractional integrals, we give the is the best possible inequality.Function   () is left continuous on [, ]; more precisely,   () is continuous and increasing on [, ] and on (, ] and has a finite jump of −1 at .Also,   () =   () = 0. Thus, we have two possibilities.
Theorem 9. Suppose that all the assumptions of Theorem 7 hold.Then, for 1 <  ≤ ∞, the following inequality holds: Proof.We apply Theorem 9 with identity function () = ,  ∈ [, ], and the proof follows directly.

Application for Hadamard Fractional Integrals
Hadamard fractional integrals of order  > 0 for 0 <  <  are given by The following result is a weighted Ostrowski type result for Hadamard fractional integrals of order .
Corollary 11.Suppose that all the assumptions of Theorem 7 and 0 <  <  hold.Then, for 1 <  ≤ ∞, the following inequality holds: The constant in the first inequality for 1 <  ≤ ∞ is sharp and the best possible for  = 1.