Power majorization between the roots of two polynomials

TL;DR

This paper establishes a mathematical relation called power majorization between roots of certain hyperbolic polynomials, confirming a numerical observation and linking polynomial factorization coefficients to root inequalities.

Contribution

It introduces a new majorization relation between roots of hyperbolic polynomials based on their quadratic factorizations, extending previous numerical findings.

Findings

Confirmed Klemeš's numerical observation

Established power majorization relation for roots

Linked polynomial coefficients to root inequalities

Abstract

It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization relation. In particular, a numerical observation by Kleme\v{s} is confirmed.