On the Koebe Quarter Theorem for Polynomials

TL;DR

This paper investigates the extremal properties of polynomials related to the Koebe Quarter Theorem, disproving a conjecture for degrees 3 to 6 and introducing new polynomial families for future conjectures.

Contribution

The paper disproves Dimitrov's conjecture for degrees 3 to 6 and introduces a new polynomial family to explore the Koebe radius problem.

Findings

Disproved the conjecture for degrees 3-6

Confirmed the conjecture for degrees 1 and 2

Proposed a new polynomial family for future research

Abstract

D. Dimitrov has posed the problem of finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal. We disprove Dimitrov's conjecture for polynomials of degree 3, 4, 5 and 6. For polynomials of degree 1 and 2 the conjecture is obviously true. On the way we introduce a new family of polynomials that allows us to state a conjecture about the value of the Koebe radius for polynomials of a specific degree.