Convexity according to means

Abstract

Given a function f : I → J and a pair of means M and N, on the intervals I and J respectively, we say that f is MN-convex provided that f(M(x, y)) ≤ N(f(x), f(y)) for every x, y ∈ I. In this context, we prove the validity of all basic inequalities in Convex Function Theory, such as Jensen's Inequality and the Hermite-Hadamard Inequality.