Characterizations and decomposition of strongly Wright-convex functions of higher order

Abstract

Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in [Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661–665] we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states that a function $f$ is strongly Wright-convex of order $n$ if and only if it is of the form $f(x)=g(x)+p(x)+c x^{n+1}$, where $g$ is a (continuous) n-convex function and $p$ is a polynomial function of degree $n$. This is a counterpart of Ng’s decomposition theorem for Wright-convex functions. We also characterize higher order strongly Wright-convex functions via generalized derivatives.