A Sandwich Theorem for Generalized Convexity and Applications

In a recent paper we have introduced the concept of (ω, t)-convexity and ω-convexity as a natural generalization of the concept of usual convexity, strong-convexity, approximate-convexity, Wright-convexity and many others. The main result of this paper gives a necessary and sufficient condition on ω under which we can separate an (ω, t)-convex (ω-convex) function and (ω, t)-concave (ω-concave) function by an (ω, t)-affine map (ω-affine map). It turns out that in both cases ω has to satisfy some functional equation. We solve these functional equations in the most frequently appearing form of the map ω. Several applications of the main results are also given.