The Hornich-Hlawka functional inequality for functions with positive differences

TL;DR

This paper explores how n-convexity influences the validity of generalized Hornich-Hlawka inequalities, extending classical results to functions with positive differences and operators.

Contribution

It extends the Hornich-Hlawka inequality to n-convex functions and positive operators, providing new insights into functional inequalities and their applications.

Findings

n-convexity ensures the inequality holds for broader classes of functions

Extensions to positive operators expand the inequality's applicability

New inequalities generalize classical results in convex analysis

Abstract

We analyze the role played by $n$-convexity for the fulfillment of a series of linear functional inequalities that extend the Hornich-Hlawka functional inequality, $f\left( x\right) +f\left( y\right) +f\left( z\right) +f\left( x+y+z\right) \geq f\left( x+y\right) +f\left( y+z\right)+f\left( z+x\right) +f(0),$ including extensions to the case of positive operators.