Superadditivity and Monotonicity of Some Functionals Associated with the Hermite-hadamard Inequality for Convex Functions in Linear Spaces

The superadditivity and monotonicity properties of some functionals associated with convex functions and the Hermite-Hadamard inequality in the general setting of linear spaces are investigated. Applications for norms and convex functions of a real variable are given. Some inequalities for arithmetic, geometric, harmonic, logarithmic and identric means are improved. 1. Introduction For any convex function we can consider the well-known inequality due to Hermite and Hadamard. It was …rst discovered by Ch. Hermite in 1881 in the journal Mathesis (see [7]). Hermite mentioned that 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: But this result was nowhere mentioned in the mathematical literature and was not widely known as Hermite’s result [8]. 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 [1]. In 1974, D.S. Mitrinovíc found Hermite’s note in Mathesis [7]. Since (1.1) was known as Hadamard’s inequality, the inequality is now commonly referred as the Hermite-Hadamard inequality [8]. Let X be a vector space, 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 [2, p. 2], [3, p. 2])