Convexity according to means

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.