# Univalent polynomials and Koebe's one-quarter theorem

## TL;DR

This paper investigates the maximum radius guaranteed by a Koebe-type theorem for univalent polynomials of fixed degree, providing a conjecture and settling cases for small degrees.

## Contribution

It formulates a conjecture on the optimal radius for univalent polynomials and confirms the conjecture for small degrees, extending classical results.

## Key findings

- Conjecture on the optimal radius for univalent polynomials.
- Verification of the conjecture for small degrees.
- Extension of Koebe's theorem to polynomial subclasses.

## Abstract

The famous Koebe $\frac14$ theorem deals with univalent (i.e., injective) analytic functions $f$ on the unit disk $\mathbb D$. It states that if $f$ is normalized so that $f(0)=0$ and $f'(0)=1$, then the image $f(\mathbb D)$ contains the disk of radius $\frac14$ about the origin, the value $\frac14$ being best possible. Now suppose $f$ is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which polynomials is it attained? A plausible conjecture is stated, and the case of small degrees is settled.