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1999/2000 Equivalence of (n+1)-th Order Peano and Usual Derivatives for n-Convex Functions
Ivan Ginchev, Angelo Guerraggio, Matteo Rocca
Real Anal. Exchange 25(2): 513-520 (1999/2000).


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.


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Ivan Ginchev. Angelo Guerraggio. Matteo Rocca. "Equivalence of (n+1)-th Order Peano and Usual Derivatives for n-Convex Functions." Real Anal. Exchange 25 (2) 513 - 520, 1999/2000.


Published: 1999/2000
First available in Project Euclid: 3 January 2009

zbMATH: 1009.26008
MathSciNet: MR1779334

Primary: 26A24

Keywords: $n$-convex functions , Peano Derivatives

Rights: Copyright © 1999 Michigan State University Press

Vol.25 • No. 2 • 1999/2000
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