论文标题
Koebe的折叠对称性定理
Koebe's theorem for trinomials with fold symmetry
论文作者
论文摘要
具有实际系数的单价多项式的KOEBE问题仅针对三项元素完全解决，这意味着在这种情况下，已经发现了Koebe半径和极端多项式（Extremiser）。总体情况保持开放，但已经提出了猜想。对于具有真实系数和$ t $折叠的旋转对称性的单价多项式，相应的猜想也已被假设。本文提供了对三项假设的确认，$ z + az^{t + 1} + bz^{2t + 1} $。也就是说，koebe半径为$ r = 4 \ cos^2 \ frac {π（1+t）} {2+3t} $，而koebe问题的唯一极端器是trinomial \ begin {chater {chater*} b^{（t）}（z）= z+\ frac2 {2+3t} \ left（-t+（2+2t）\ cos \ cos \ frac {πt} {2+3t} {2+3t} \ right）z^{1+t}+t} +\ frac1 {2+3t} \ left（2+t-2t \ cos \ frac {πt} {2+3t} \ right）z^{1+2t}。 \ end {收集*} 关键词和短语：Koebe四分之一定理，Koebe半径，单价多项式，具有折叠对称性的三项式。
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 $r=4\cos^2 \frac{π(1+T)}{2+3T}$, 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{πT}{2+3T}\right)z^{1+T}+\\ +\frac1{2+3T}\left(2+T-2T\cos\frac{πT}{2+3T}\right)z^{1+2T}. \end{gather*} Key words and phrases: Koebe one-quarter theorem, Koebe radius, univalent polynomial, trinomials with fold symmetry.
