On the Koebe Quarter Theorem for Polynomials
| dc.contributor.author | Олександр Михайлович Стоколос, Jimmy Dillies, Dmitriy Dmitrishin, Andrey Smorodin | |
| dc.date.accessioned | 2023-05-10T13:22:47Z | |
| dc.date.available | 2023-05-10T13:22:47Z | |
| dc.date.issued | 2022 | |
| dc.description.abstract | 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. | |
| dc.identifier.issn | 2409-8906 | |
| dc.identifier.uri | https://card-file.ontu.edu.ua/handle/123456789/24993 | |
| dc.source | Proceedings of the International Geometry Center | |
| dc.title | On the Koebe Quarter Theorem for Polynomials |