(k,s)-Riemann-Liouville fractional integral and applications Mehmet Zeki SARIKAYA ∗† , Zoubir DAHMANI ‡ , Mehmet Ey¨ up KIRIS § and Farooq AHMAD ¶ Abstract In this paper, we introduce a new approach on fractional integration, which generalizes the Riemann-Liouville fractional integral. We prove some properties for this new approach. We also establish some new integral inequalities using this new fractional integration. Keywords: Riemann-Liouville fractional integrals, synchronous function, Cheby- shev inequality, H¨older inequality. 2000 AMS Classification: 26A33;26A51;26D15. 1. Introduction Fractional calculus and its widely application have recently been paid more and more attentions. For more recent development on fractional calculus, we refer the reader to [7, 12, 15, 16, 19]. There are several known forms of the fractional integrals of which two have been studied extensively for their applications [5, 10, 11, 14, 21]. The first is the Riemann-Liouville fractional integral of α ≥ 0 for a continuous function f on [a, b] which is defined by J α a f (x)= 1 Γ(α)  x a (x − t) α-1 f (t)dt, α ≥ 0,a<x ≤ b. This integral is motivated by the well known Cauchy formula:  x a dt 1  t1 a dt 2 ...  tn-1 a f (t n )dt n = 1 Γ(n)  x a (x − t) n-1 f (t)dt, n ∈ N * . The second is the Hadamard fractional integral introduced by Hadamard [9]. It is given by: J α a f (x)= 1 Γ(α)  x a  log x t  α-1 f (t) dt t , α> 0, x > a. * Department of Mathematics, Faculty of Science and Arts, D¨ uzce University, D¨ uzce, Turkey, Email: sarikayamz@gmail.com † Corresponding Author. ‡ Laboratory of Pure and Applied Mathematics, UMAB, University of Mostaganem, Algeria, Email: zzdahmani@yahoo.fr § Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon-TURKEY, Email: mkiris@gmail.com, kiris@aku.edu.tr ¶ Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya Uni- versity, Multan, 60800, Pakistan, Email: farooqgujar@gmail.com