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

The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s has all roots in the open upper half-plane if and only if s=p+iq and q and k-1 has a negative leading coefficient. Considering roots of p \mathbb{R}P^1 \mathbb{C} P^1 as its diameter is the maximal univalent disk for the function R=\frac{q}{p}