Symmetric polynomials and $l^p$ inequalities for certain intervals of $p$

TL;DR

This paper establishes new, weaker conditions based on symmetric polynomials that guarantee $l^p$ inequalities between vectors over specific $p$ intervals, extending the understanding beyond traditional majorization.

Contribution

It introduces novel sufficient conditions using symmetric polynomials for $l^p$ inequalities that are weaker than majorization, and characterizes majorization via these polynomials.

Findings

Derived weaker conditions for $l^p$ inequalities using symmetric polynomials.

Characterized majorization in terms of symmetric polynomials.

Extended the range of $p$ for which inequalities hold.

Abstract

We prove some sufficient conditions implying $l^p$ inequalities of the form $||x||_p \leq ||y||_p$ for vectors $ x, y \in [0,\infty)^n$ and for $p$ in certain positive real intervals. Our sufficient conditions are strictly weaker than the usual majorization relation. The conditions are expressed in terms of certain homogeneous symmetric polynomials in the entries of the vectors. These polynomials include the elementary symmetric polynomials as a special case. We also give a characterization of the majorization relation by means of symmetric polynomials.