Equivalence of \((n+1)\)-th order Peano and usual derivatives for \(n\)-convex functions. (English) Zbl 1009.26008

Ginchev, Ivan; Guerraggio, Angelo; Rocca, Matteo

Real Anal. Exch. 25(1999-2000), No. 2, 513-520 (2000).

Summary: 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, \(f^{(r)}_+\), \(1\leq r\leq n\), 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.

MSC:

26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A51	Convexity of real functions in one variable, generalizations

Keywords:

\(n\)-convex functions; right derivative; Peano derivative