On the Koebe quarter theorem for polynomials

Jimmy Dillies, Dmitriy Dmitrishin, Andrey Smorodin, Alex Stokolos

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 [8] 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 [4] and D. Dimitrov [7]. 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 51. We then discuss the ensuing estimates for the value of the Koebe radius for polynomials of a specific degree.

Original languageEnglish
Pages (from-to)219-230
Number of pages12
JournalProceedings of the International Geometry Center
Volume14
Issue number3
DOIs
StatePublished - 2021

Keywords

  • Koebe one-quarter theorem
  • Koebe radius
  • Univalent polynomial

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