Symmetric polynomials, p-norm inequalities, and certain functionals related to majorization

TL;DR

This paper investigates a modified majorization relation involving symmetric polynomials, explores related inequalities, and discusses their connection to longstanding conjectures on Lp inequalities for exponential sums.

Contribution

It introduces a variant of majorization involving Schur-concave polynomials and analyzes its implications for classical conjectures in harmonic analysis.

Findings

Identifies limitations of usual majorization in Lp inequality contexts

Proposes a new inequality framework involving symmetric polynomials

Provides insights into conjectures by Hardy and Littlewood

Abstract

We study a variant of the majorization relation. In particular we consider inequalities involving some Schur-concave symmetric polynomials related to the multinomial expansion. We also discuss how these topics were motivated by conjectures on sharp Lp inequalities between complex exponential sums conjectured by Hardy and Littlewood (still open problems), and why the usual majorization relation does not hold in that context.