English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Koebe problem for univalent polynomials with real coefficients is fullysolved only for trinomials, which means that in this case the Koebe radius andthe extremal polynomial (extremizer) have been found. The general case remainsopen, but conjectures have been formulated. The corresponding conjectures havealso been hypothesized for univalent polynomials with real coefficients and$T$-fold rotational symmetry. This paper provides confirmation of thesehypotheses for trinomials $z + az^{T + 1} + bz^{2T + 1}$. Namely, the Koeberadius is $r=4\cos^2 \frac{\pi(1+T)}{2+3T}$, and the only extremizer of theKoebe 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, univalentpolynomial, trinomials with fold symmetry.

Koebe's theorem for trinomials with fold symmetry