The hierarchy of convexity and some classic inequalities

Abstract

In what follows, a hierarchy of m-convexity is considered: we define m-starshaped functions, m-superadditive functions, Jensen m-convex functions, weak Jensen m-convex func- tions, Jensen m-superadditive functions, and weak m-superadditive functions. Some inclusions between such classes of functions are established. We also analyze the validity of the Hermite- Hadamard inequality, and of the Chebyshev-Andersson inequality for m-convex functions.