Maximal univalent disks of real rational functions and Hermite-Biehler polynomials

TL;DR

This paper characterizes the maximal univalent disks of certain rational functions derived from Hermite-Biehler polynomials, providing a solution to a special case of the Hermite-Biehler problem.

Contribution

It identifies the maximal univalent disk for the function R=q/p where p and q satisfy Hermite-Biehler conditions, solving a specific case of the Hermite-Biehler problem.

Findings

The open disk with a pair of consecutive roots of p as diameter is maximal univalent for R=q/p.

The result applies to polynomials with real, simple, interlacing roots.

It advances understanding of univalent regions for rational functions related to Hermite-Biehler polynomials.

Abstract

The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k has all roots in the open upper half-plane if and only if s=p+iq where p and q are real polynomials of degree k and k-1 resp. with all real, simple and interlacing roots, and q has a negative leading coefficient. Considering roots of p as cyclically ordered on RP^1 we show that the open disk in CP^1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R=\frac{q}{p}. This solves a special case of the so-called Hermite-Biehler problem.