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

We present Montgomery identity for Riemann-Liouville fractional integral as well as for fractional integral of a function with respect to another function . We further use them to obtain Ostrowski type inequalities involving functions whose first derivatives belong to spaces. These inequalities are generally sharp in case and the best possible in case . Application for Hadamard fractional integrals is given. 1. Introduction The following Ostrowski inequality is well known [1]: It holds for every whenever is continuous on and differentiable on with derivative bounded on ; that is, 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 is defined by where is a gamma function When (5) is the Riemann definition of fractional integral. In case , fractional integral reduces to classical integral. In [3], the following Montgomery identity for fractional integrals is obtained. Theorem 1. Let , differentiable on , integrable on , and . Then, the following identity holds: where In [3], the authors used this identity to obtain the following Ostrowski type inequality for fractional integrals. Theorem 2. Let , differentiable on , and for every ; then, the following Ostrowski inequality holds: 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, . 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 and the best possible in case . As a special case, application for Hadamard fractional integrals is given. 2. Montgomery Identity for Fractional Integrals In this section we give another, simpler new generalization of