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

Authors: Vladimir P. Kostov, Boris Shapiro and Mikhail Tyaglov
Journal: Proc. Amer. Math. Soc. 139 (2011), 1625-1635
MSC (2010): Primary 26C05; Secondary 30C15
Published electronically: November 4, 2010
MathSciNet review: 2763752
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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$ 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.

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Additional Information

Vladimir P. Kostov
Affiliation: Laboratoire de Mathématiques, Université de Nice, Parc Valrose, 06108 Nice Cedex 2, France

Boris Shapiro
Affiliation: Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden

Mikhail Tyaglov
Affiliation: Institut für Mathematik, MA 4-5 Technische Universität Berlin, D-10623 Berlin, Germany

Keywords: Hermite-Biehler theorem, root localization
Received by editor(s): May 4, 2010
Published electronically: November 4, 2010
Additional Notes: The third author was supported by the Sofja Kovalevskaja Research Prize of the Alexander von Humboldt Foundation.
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society