INEQUALITIES OF HERMITE-HADAMARD TYPE FOR h-CONVEX FUNCTIONS ON LINEAR SPACES

Some inequalities of Hermite-Hadamard type for h-convex functions de…ned on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well. 1. Introduction The following inequality holds for any convex function f de…ned on R (1.1) (b a)f a+ b 2 < Z b a f(x)dx < (b a) + f(b) 2 ; a; b 2 R: It was …rstly discovered by Ch. Hermite in 1881 in the journal Mathesis (see [41]). But this result was nowhere mentioned in the mathematical literature and was not widely known as Hermite’s result. E. F. Beckenbach, a leading expert on the history and the theory of convex functions, wrote that this inequality was proven by J. Hadamard in 1893 [5]. In 1974, D. S. Mitrinovíc found Hermite’s note in Mathesis [41]. Since (1.1) was known as Hadamard’s inequality, the inequality is now commonly referred as the Hermite-Hadamard inequality. For related results, see [10]-[19], [22]-[24], [31]-[34] and [44]. Let X be a vector space over the real or complex number …eld K and x; y 2 X; x 6= y. De…ne the segment [x; y] := f(1 t)x+ ty; t 2 [0; 1]g: We consider the function f : [x; y]! R and the associated function g(x; y) : [0; 1]! R; g(x; y)(t) := f [(1 t)x+ ty]; t 2 [0; 1]: Note that f is convex on [x; y] if and only if g(x; y) is convex on [0; 1]. For any convex function de…ned on a segment [x; y] X, we have the HermiteHadamard integral inequality (see [20, p. 2], [21, p. 2])