New integral representations of nth order convex functions

In this paper we give an integral representation of an n-convex function f in general case without additional assumptions on function f. We prove that any n-convex function can be represented as a sum of two ( n + 1 ) -times monotone functions and a polynomial of degree at most n. We obtain a decomposition of n-Wright-convex functions which generalizes and complements results of Maksa and Páles (2009) [13]. We define and study relative n-convexity of n-convex functions. We introduce a measure of n-convexity of f. We give a characterization of relative n-convexity in terms of this measure, as well as in terms of nth order distributional derivatives and Radon–Nikodym derivatives. We define, study and give a characterization of strong n-convexity of an n-convex function f in terms of its derivative f ( n + 1 ) ( x ) (which exists a.e.) without additional assumptions on differentiability of f. We prove that for any two n-convex functions f and g, such that f is n-convex with respect to g, the function g is the support for the function f in the sense introduced by Wąsowicz (2007) [29], up to polynomial of degree at most n.