An Extremal Problem for Odd Univalent Polynomials

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*}

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An Extremal Problem for Odd Univalent Polynomials

Cited by 2 publications

References 11 publications

Extremal problems for typically real odd polynomials

Extremal problems for typically real odd polynomials

On the Koebe quarter theorem for trinomials with fold symmetry

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