Pak. J. Engg. & Appl. Sci. Vol. 13, July., 2013 (p. 127-133) 127 New Inequalities of Hadamard-type via -Convexity Muhammad Iqbal 1* , M. Iqbal Bhatti 1 and S. Hussain 1 1. Department of Mathmnematics, University of Engineering & Technology, Lahore. * Corresponding Author: miqbal.bki@gmail.com Abstract Some new Hermite-Hadamard’s inequalities for h-convex functions are proved, generalizing some results in [1, 3, 6] and unifying a number of known results. Some new applications for special means of real numbers are also deduced. Key Words: Hermite-Hadamard inequality, Hölder’s inequality, Convex and h-convex functions. 1. Introduction and Preliminaries If a function   ] , [ : b a f is convex, then         b a b f a f a b b a dx x f f 2 ) ( ) ( 1 2 ) ( (1) is known as the Hermite-Hadamard inequality. For concave function f, the above order is reversed. Inequality (1) is refined, extended, generalized and new proofs are given in [1, 2, 3, 5, 7, 8]. Now we present definitions, theorems and results that we apply in this paper. Definition 1. [1] Let I be an interval of real numbers. A function l f : is said to be convex if for all l y x  , and  t [0, 1] ) ( ) 1 ( ) ( } ) 1 ( { y f t x tf y t tx f      f is said to be concave, if the above inequality is reversed. Definition 2. [4] A non-negative function  l f :   is said to Godunova-Levin function (or f is said to belong to class Q(l)) if, f or all x, y  l and t  (0, 1)   t y f t x f y t tx f      1 ) ( ) ( ) 1 ( It may be noted that this class contained all non- negative monotone and non-negative convex functions . Definition 3. [2] A function ) , 0 [ ) , 0 [ :    f is a function of P type (or that f belongs to the class P(l)) if, for all x, y  [0, ) and t  [0, 1]   ) ( ) ( ) 1 ( y f x f y t tx f     Definition 4: [1, p.288] A function   ) , 0 [ : f is said to be s-convex function in the second sense (or ) 2 S K f  if for all x, y  [0, ), t [0, 1] and s  [0,1], the following inequality holds:   ) ) ) 1 ( ) ( ) 1 ( y f t x f t y t tx f s s      Obviously, 1-convex function is convex. Definition 5 [10] Let I, J be intervals in J   ) 1 , 0 ( , and let  J h :   be a non-negative function, 0  h . A non-negative function f:l  is called h-convex function (or f belongs to the class SX, (h, l)), if for all x, y  l and t  (0, 1), the inequality   ) ( ) 1 ( ) ( ) ( ) 1 ( y f t h x f t h y t tx f      holds If the inequality is reversed then f is said to be h- concave and in this case f belongs to the class SV(h,l). Remark 1. If h(t) = t, then all the non-negative convex functions belong to the class SX(h, l) and all non-negative concave functions belong to the class SV (h, l). If , 1 ) ( t t h  then SX(h; l) = Q(l). If , 1 ) (  t h then SX(h; l)  P(l). If h(t) = t s , where s  (0, 1), then SX (h, l)  2 s K In [8] some new Hadamard-type inequalities for h-convex functions are discussed by authors.