On Some Generalized Integral Inequalities for Riemann-Liouville Fractional Integrals. Mehmet Zeki Sarikayaa, Hatice Filiza, Mehmet Ey ¨up Kirisb. aDepartment ...
Filomat 29:6 (2015), 1307–1314 DOI 10.2298/FIL1506307S
Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat
On Some Generalized Integral Inequalities for Riemann-Liouville Fractional Integrals Mehmet Zeki Sarikayaa , Hatice Filiza , Mehmet Eyup ¨ Kirisb a Department b Department
of Mathematics, Faculty of Science and Arts, Duzce ¨ University, Duzce, ¨ Turkey of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon, Turkey
Abstract. In this paper, we give a generalized Montogomery identities for the Riemann-Liouville fractional integrals. We also use this Montogomery identities to establish some new Ostrowski type integral inequalities.
1. Introduction The inequality of Ostrowski [20] 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 Zb a+b 2
1 1 (x − 2 ) f (x) −
f 0 f (t)dt ≤ + (b − a) ∞ b−a 4 (b − a)2 a
for every x ∈ [a, b]. Moreover the constant 1/4 is the best possible. For some generalizations of this classic fact see the book [9, p.468-484] by Mitrinovic, Pecaric and Fink. A simple proof of this fact can be done by using the following identity [9]: If f : [a, b] → R is differentiable on [a, b] with the first derivative f 0 integrable on [a, b], then Montgomery identity holds: 1 f (x) = b−a
Zb
Zb f (t)dt +
a
P1 (x, t) f 0 (t)dt, a
where P1 (x, t) is the Peano kernel defined by t−a , a≤t