On the Koebe Quarter Theorem for Polynomials
Complex Variables
2019-04-26 v1 Classical Analysis and ODEs
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.
Cite
@article{arxiv.1904.11039,
title = {On the Koebe Quarter Theorem for Polynomials},
author = {Jimmy Dillies and Dmitriy Dmitrishin and Andrey Smorodin and Alex Stokolos},
journal= {arXiv preprint arXiv:1904.11039},
year = {2019}
}