On the Koebe quarter theorem for trinomials with fold 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 T -fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials z + a z T + 1 + b z 2 T + 1 z + az^{T + 1} + bz^{2T + 1} . Namely, the Koebe radius is r = 4 cos 2 ⁡ ( π ( 1 + T ) 2 + 3 T ) r=4\cos ^2\left (\frac {\pi (1+T)}{2+3T}\right ) , and the only extremizer of the Koebe problem is the trinomial B ( T ) ( z ) = z + 2 2 + 3 T ( − T + ( 2 + 2 T ) cos ⁡ ( π T 2 + 3 T ) ) z 1 + T + + 1 2 + 3 T ( 2 + T − 2 T cos ⁡ ( π T 2 + 3 T ) ) z 1 + 2 T . \begin{gather*} B^{(T)}(z)=z+\frac 2{2+3T}\left (-T+(2+2T)\cos \left (\frac {\pi T}{2+3T}\right )\right )z^{1+T}+\\ +\frac 1{2+3T}\left (2+T-2T\cos \left (\frac {\pi T}{2+3T}\right )\right )z^{1+2T}. \end{gather*}