Journal of Inequalities and Special Functions ISSN: 2217-4303, URL: http://ilirias.com/jiasf Volume 7 Issue 4(2016), Pages 150-167. ON HADAMARD TYPE INEQUALITIES FOR m-CONVEX FUNCTIONS VIA FRACTIONAL INTEGRALS G. FARID 1 , A. UR REHMAN 2 , B. TARIQ 3 , AND A. WAHEED 4 Abstract. In this paper, we prove the Hadamard type inequalities for m- convex functions via fractional integrals and related inequalities. These results have some relationships with the Hadamard inequalities for fractional integrals and related inequalities. 1. Introduction The Hadamard inequality states that: If f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a<b, then f  a + b 2  ≤ 1 b - a  b a f (x)dx ≤ f (a)+ f (b) 2 . The Hadamard inequality have got attention of many mathematicians and many generalizations and refinements have been found so far for example see, [2, 3, 4, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 22, 23]. In [24] Toader define the concept of m-convexity, an intermediate between usual convexity and star shape functions. Definition 1.1. A function f : [0,b] → R, b> 0, is said to be m-convex, where m ∈ [0, 1], if we have f (tx + m(1 - t)y) ≤ tf (x)+ m(1 - t)f (y) for all x, y ∈ [0,b] and t ∈ [0, 1]. If we take m = 1, then we recapture the concept of convex functions defined on [0,b] and if we take m = 0, then we get the concept of starshaped functions on [0,b]. We recall that f : [0,b] → R is called starshaped if f (tx) ≤ tf (x) for all t ∈ [0,b] and x ∈ [0,b]. Denote by K m (b) the set of the m-convex functions on [0,b] for which f (0) < 0, then one has K 1 (b) ⊂ K m (b) ⊂ K 0 (b), 2000 Mathematics Subject Classification. 26B15, 26A51, 34L15. Key words and phrases. Hadamard inequality, convex functions, Riemann-Liouville fractional integrals. c 2016 Universiteti i Prishtin¨ es, Prishtin¨ e, Kosov¨ e. Submitted July 21, 2016. Published November 11, 2016. 150