Characterizations of higher-order convexity properties with respect to Chebyshev systems

TL;DR

This paper introduces and characterizes various higher-order convexity notions for real functions with respect to Chebyshev systems, using generalized divided differences and Dinghas derivatives.

Contribution

It provides new characterizations of convexity notions via generalized divided differences and derivatives related to Chebyshev systems, extending existing convexity theory.

Findings

Convexity notions characterized by nonnegativity of generalized divided differences.

Introduction of a relevant divided difference and Dinghas type derivative.

Main results connect convexity with these generalized derivatives.

Abstract

In this paper various notions of convexity of real functions with respect to Chebyshev systems defined over arbitrary subsets of the real line are introduced. As an auxiliary notion, a concept of a relevant divided difference and also a related lower Dinghas type derivative are also defined. The main results of the paper offer various characterizations of the convexity notions in terms of the nonnegativity of a generalized divided difference and the corresponding lower Dinghas type derivative.