## 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 language | English |
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Pages (from-to) | 219-230 |

Number of pages | 12 |

Journal | Proceedings of the International Geometry Center |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - 2021 |

## Keywords

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