Inequalities from Lorentz-Finsler norms

TL;DR

This paper demonstrates how Lorentz-Finsler geometry can be used to derive and refine various classical inequalities, including new versions of Aczél's inequality, by unifying them under a geometric framework.

Contribution

It introduces a novel geometric approach using Lorentz-Finsler norms to derive and refine classical inequalities, revealing their underlying geometric structure.

Findings

Classical inequalities are special cases of Lorentz-Finsler reverse inequalities

New refinements of Aczél's inequality are established

Lorentz-Finsler geometry provides a unifying framework for inequalities

Abstract

We show that Lorentz-Finsler geometry offers a powerful tool in obtaining inequalities. With this aim, we first point out that a series of famous inequalities such as: the (weighted) arithmetic-geometric mean inequality, Acz\'el's, Popoviciu's and Bellman's inequalities, are all particular cases of a reverse Cauchy-Schwarz, respectively, of a reverse triangle inequality holding in Lorentz-Finsler geometry. Then, we use the same method to prove some completely new inequalities, including two refinements of Acz\'el's inequality.