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

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

2010-05-10

arxiv.org/abs/1005.1596v1

Proc. Amer. Math. Soc. 139 (2011), no. 5, pp. 1625-1635

Mathematics

Classical Analysis and ODEs

10 pages, 4 figures

Scientific paper

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.

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