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

Kostov, Vladimir; Shapiro, Boris; Tyaglov, Mikhail

doi:10.1090/S0002-9939-2010-10778-5

Maximal univalent disks of real rational functions and Hermite-Biehler polynomials
HTML articles powered by AMS MathViewer

by Vladimir P. Kostov, Boris Shapiro and Mikhail Tyaglov PDF

Proc. Amer. Math. Soc. 139 (2011), 1625-1635 Request permission

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

Jean-Pierre Dedieu, Obreschkoff’s theorem revisited: what convex sets are contained in the set of hyperbolic polynomials?, J. Pure Appl. Algebra 81 (1992), no. 3, 269–278. MR 1179101, DOI 10.1016/0022-4049(92)90060-S
Karl Dilcher and Kenneth B. Stolarsky, Zeros of the Wronskian of a polynomial, J. Math. Anal. Appl. 162 (1991), no. 2, 430–451. MR 1137630, DOI 10.1016/0022-247X(91)90160-2
A. Eremenko and A. Gabrielov, Degrees of real Wronski maps, Discrete Comput. Geom. 28 (2002), no. 3, 331–347. MR 1923956, DOI 10.1007/s00454-002-0735-x
Alexandre Eremenko and Andrei Gabrielov, The Wronski map and Grassmannians of real codimension 2 subspaces, Comput. Methods Funct. Theory 1 (2001), no. 1, [On table of contents: 2002], 1–25. MR 1931599, DOI 10.1007/BF03320973
S. Fisk, Polynomials, roots, and interlacing, arXiv:math/0612833.
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
Zeev Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551. MR 29999, DOI 10.1090/S0002-9904-1949-09241-8
Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002. MR 1954841
Saeed Zakeri, On critical points of proper holomorphic maps on the unit disk, Bull. London Math. Soc. 30 (1998), no. 1, 62–66. MR 1479037, DOI 10.1112/S0024609397003706