Moduli of roots of hyperbolic polynomials and Descartes' rule of signs

Abstract

A real univariate polynomial with all roots real is called hyperbolic. By Descartes' rule of signs for hyperbolic polynomials (HPs) with all coefficients nonvanishing, a HP with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients has exactly $c$ positive and $p$ negative roots. For $c=2$ and for degree $6$ HPs, we discuss the question: When the moduli of the $6$ roots of a HP are arranged in the increasing order on the real half-line, at which positions can be the moduli of its two positive roots depending on the positions of the two sign changes in the sequence of coefficients?