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MAXIMAL UNIVALENT DISKS OF REAL RATIONAL FUNCTIONS AND HERMITE-BIEHLER POLYNOMIALS

By Vladimir P. Kostov, Boris Shapiro, Mikhail Tyaglov and Communicated Ken Ono

Abstract

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, wherepand 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 RP 1 we show that the open disk in CP 1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R = q. This solves a p special case of the so-called Hermite-Biehler problem. 1

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.3720
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