Open AccessOn the Koebe Quarter Theorem for Polynomials
Author(s) -
Олександр Михайлович Стоколос,
Jimmy Dillies,
Dmitriy Dmitrishin,
Andrey Smorodin
Publication year - 2022
Publication title -
trudy meždunarodnogo geometričeskogo centra/pracì mìžnarodnogo geometričnogo centru
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.124
H-Index - 2
eISSN - 2409-8906
pISSN - 2072-9812
DOI - 10.15673/tmgc.v14i3.2057
Subject(s) - mathematics , radius , degree (music) , value (mathematics) , conjecture , quarter (canadian coin) , function (biology) , combinatorics , pure mathematics , computer science , statistics , physics , computer security , archaeology , evolutionary biology , biology , acoustics , history
The Koebe One Quarter Theorem states that the range of any Schlicht function contains the centered disc of radius 1/4 which is sharp due to the value of the Koebe function at −1. A natural question is finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials. In particular, it was asked in [7] whether Suffridge polynomials [15] are optimal. For polynomials of degree 1 and 2 that is obviously true. It was demonstrated in [10] that Suffridge polynomials of degree 3 are not optimal and a promising alternative family of polynomials was introduced. These very polynomials were actually discovered earlier independently by M. Brandt [3] and D. Dimitrov [9]. In the current article we reintroduce these polynomials in a natural way and make a far-reaching conjecture that we verify for polynomials up to degree 6 and with computer aided proof up to degree 52. We then discuss the ensuing estimates for the value of the Koebe radius for polynomials of a specific degree.