On the Structure of (s,t)-Convex Functions

Abstract

A function f: I→[-∞,∞[(I⊂ℝ interval) is said to be (s,t)-convex (for fixed s,t∈]0,1[) iff one has f(su+(1-s)v)≤tf(u)+(1-t)f(v) for all u,v∈i. We prove that such a function is necessarily (t,t)-convex for all rational tε]0,1[. Furthermore we show that a function which fulfills the (s,t)-convexity inequality in a weakened sense is closely related to a uniquely determined (s,t)-convex function.