Ohlin’s lemma and some inequalities of the Hermite–Hadamard type

Abstract

Using the Ohlin lemma on convex stochastic ordering we prove inequalities of the Hermite–Hadamard type. Namely, we determine all numbers a,α,β∈[0,1] such that for all convex functions f the inequality af(αx+(1−α)y)+(1−a)f(βx+(1−β)y)≤1y−x∫xyf(t)dt is satisfied and all a,b,c,α∈(0,1) with a + b + c = 1 for which we have af(x)+bf(αx+(1−α)y)+cf(y)≥1y−x∫xyf(t)dt