Maximal univalent disks of real rational functions and 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 $k-1$ respectively with all real, simple and interlacing roots, and $q$ has a negative leading coefficient. Considering roots of $p$ as cyclically ordered on $\mathbb {R}P^1$ we show that the open disk in $\mathbb {C} P^1$ 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.