Abstract
We present Montgomery identity for Riemann-Liouville fractional integral as well as for fractional integral of a functionfwith respect to another functiong. We further use them to obtain Ostrowski type inequalities involving functions whose first derivatives belong toLpspaces. These inequalities are generally sharp in casep>1and the best possible in casep=1. Application for Hadamard fractional integrals is given.
Highlights
The following Ostrowski inequality is well known [1]: f (x) − b 1 − a b ∫ a f (t) dt (1) ≤ [ +(x ((a + b) /2))2 (b − a
We give another, simpler new generalization of Montgomery identity for Riemann-Liouville fractional integral of order α, which holds for a larger set of α; that is, α > 0
The fractional integral of a function f with respect to another function g is given by aJαx;gf (x)
Summary
It holds for every x ∈ [a, b] whenever f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) with derivative f : (a, b) → R bounded on (a, b); that is, Ostrowski proved this inequality in 1938, and since 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 be proved by using the following Montgomery identity (see, for instance, [2]):. We give another, simpler new generalization of Montgomery identity for Riemann-Liouville fractional integral of order α, which holds for a larger set of α; that is, α > 0. Application for Hadamard fractional integrals is given
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