On Some Generalized Integral Inequalities for Riemann-Liouville ...

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