Implications between generalized convexity properties of real functions

Motivated by the well-known implications among t-convexity properties of real functions, analogous relations among the upper and lower M-convexity properties of real functions are established. More precisely, having an n-tuple (M1,…,Mn) of continuous two-variable means, the notion of the descendant of these means (which is also an n-tuple (N1,…,Nn) of two-variable means) is introduced. In particular, when all the means Mi are weighted arithmetic, then the components of their descendants are also weighted arithmetic means. More general statements are obtained in terms of the generalized quasi-arithmetic or Matkowski means. The main results then state that if a function f is Mi-convex for all i∈{1,…,n}, then it is also Ni-convex for all i∈{1,…,n}. Several consequences are discussed.