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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Maximal univalent disks of real rational functions and Hermite-Biehler polynomials
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by Vladimir P. Kostov, Boris Shapiro and Mikhail Tyaglov PDF
Proc. Amer. Math. Soc. 139 (2011), 1625-1635 Request permission


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:
  • Boris Shapiro
  • Affiliation: Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden
  • MR Author ID: 212628
  • Email:
  • Mikhail Tyaglov
  • Affiliation: Institut für Mathematik, MA 4-5 Technische Universität Berlin, D-10623 Berlin, Germany
  • Email:
  • 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
  • © Copyright 2010 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 1625-1635
  • MSC (2010): Primary 26C05; Secondary 30C15
  • DOI:
  • MathSciNet review: 2763752