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arXiv · 1005.1596

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

V. Kostov·B. Shapiro·M. Tyaglov
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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.

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V. Kostov, B. Shapiro, M. Tyaglov. 2010-05-10. Maximal univalent disks of real rational functions and Hermite-Biehler polynomials. https://doi.org/10.1090/s0002-9939-2010-10778-5

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