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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
DOI: https://doi.org/10.1090/S0002-9939-2010-10778-5
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
Email: kostov@unice.fr

Boris Shapiro
Affiliation: Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden
MR Author ID: 212628
Email: shapiro@math.su.se

Mikhail Tyaglov
Affiliation: Institut für Mathematik, MA 4-5 Technische Universität Berlin, D-10623 Berlin, Germany
Email: tyaglov@math.tu-berlin.de

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