On the classes of higher-order Jensen-convex functions and Wright-convex functions, II

TL;DR

This paper resolves an open problem by proving that for all natural numbers n, the class of n-Wright-convex functions is a proper subset of n-Jensen-convex functions, extending previous results to even n.

Contribution

It provides a complete proof that the inclusion of n-Wright-convex functions in n-Jensen-convex functions is proper for all n, including even cases, and compares strongly convex classes.

Findings

The inclusion is proper for all natural n.

The result extends to strongly convex function classes.

Completes the classification for even n cases.

Abstract

Recently Nikodem, Rajba and W\k{a}sowicz compared the classes of n-Wright-convex functions and n-Jensen-convex functions by showing that the first one is a proper subclass of the latter one, whenever n is an odd natural number. Till now the case of even n was an open problem. In this paper the complete solution is given: it is shown that the inclusion is proper for any natural n. The classes of strongly n-Wright-convex and strongly n-Jensen-convex functions are also compared (with the same assertion).