Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 26C05, 30C15

Retrieve articles in all journals with MSC (2010): 26C05, 30C15


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
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10778-5
PII: S 0002-9939(2010)10778-5
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