Koebe's theorem for trinomials with fold symmetry

TL;DR

This paper confirms conjectures about the Koebe radius and extremal polynomials for trinomials with fold symmetry, extending known results to a broader class of univalent polynomials with specific symmetry properties.

Contribution

The paper proves the Koebe radius and identifies the extremizer for a class of trinomials with fold symmetry, confirming previous conjectures in this area.

Findings

Koebe radius for these trinomials is explicitly calculated.

The extremal polynomial for the Koebe problem is explicitly determined.

Results extend known cases to polynomials with T-fold rotational symmetry.

Abstract

The Koebe problem for univalent polynomials with real coefficients is fully solved only for trinomials, which means that in this case the Koebe radius and the extremal polynomial (extremizer) have been found. The general case remains open, but conjectures have been formulated. The corresponding conjectures have also been hypothesized for univalent polynomials with real coefficients and $T$-fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials $z + az^{T + 1} + bz^{2T + 1}$. Namely, the Koebe radius is $r=4\cos^2 \frac{\pi(1+T)}{2+3T}$, and the only extremizer of the Koebe problem is the trinomial \begin{gather*} B^{(T)}(z)=z+\frac2{2+3T}\left(-T+(2+2T)\cos\frac{\pi T}{2+3T}\right)z^{1+T}+\\ +\frac1{2+3T}\left(2+T-2T\cos\frac{\pi T}{2+3T}\right)z^{1+2T}. \end{gather*} Key words and phrases: Koebe one-quarter theorem, Koebe radius, univalent polynomial, trinomials with fold symmetry.