Equivalence of (n+1)-th Order Peano and Usual Derivatives for n-Convex Functions

Abstract

A real-valued function $f$ defined on an interval of $\mathbb{R}$ is said to be $n$-convex if all its $n$-th order divided differences are not negative. Let $f$ be such a function defined in a right neighborhood of $t_0 \in \mathbb{R}$ whose usual right derivatives, $f^{(r)}_+,$ $1\le r\le n,$ exist in that neighborhood and whose $(n+1)$-th order Peano derivative, $f_{n+1}(t_0)$, exists at $t_0$. Under these assumptions we prove that $f$ also possesses $(n+1)$-th order usual right derivative $f^{(n+1)}_+(t_0)$ at $t_0$. This result generalizes the known case for convex (that is 1-convex) functions. The latter appears in works of B. Jessen studying the curvature of convex curves and of J. M. Borwein, M. Fabian, D. Noll studying the second order differentiability of convex functions on abstract spaces.