Descartes' rule of signs and moduli of roots

Abstract

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. For $c=1$ and $2$, we discuss the question: When the moduli of all the roots of a HP are arranged in the increasing order on the real half-line, at which positions can be the moduli of its positive roots depending on the positions of the sign changes in the sequence of coefficients?