Majorization Inequalities from Logarithmic Convexity

TL;DR

This paper introduces a unifying approach using log-convexity to establish majorization inequalities for symmetric polynomials, resolving longstanding conjectures and extending classical results.

Contribution

It demonstrates that log-convexity serves as a versatile tool to prove majorization inequalities, unifying and extending previous results for various classes of polynomials.

Findings

Proved new majorization inequalities for Macdonald, Jack, and Heckman-Opdam hypergeometric functions.

Unified existing results and resolved several open conjectures.

Showed that log-convexity implies majorization and is preserved under key operations.

Abstract

Majorization inequalities for symmetric polynomials have interested mathematicians for centuries, from the AM-GM inequality for two variables going back at least to Euclid, through classical results of Newton, Muirhead and Gantmacher, to more recent extensions to Schur polynomials and zonal spherical functions. These have been established case by case, with no unified approach. Although it is known that majorization inequalities follow from symmetry and convexity in the indexing partition, the difficulty of proving convexity in specific cases has left a number of outstanding conjectures inaccessible until now. The key insight of this paper is that log-convexity provides both a more versatile tool and a unifying principle. It implies convexity and hence majorization, and it is preserved under multiplication and weighted averaging, making it well suited to inductive arguments in a wide range of settings. Using this idea, we prove new majorization inequalities for Macdonald polynomials, Jack polynomials and Heckman-Opdam hypergeometric functions, unifying existing results and resolving several open conjectures.