An extremal problem for odd univalent polynomials

For the univalent polynomials $F(z) = \sum\limits_{j=1}^{N} a_j z^{2j-1}$ with real coefficients and normalization $a_1 = 1$ we solve the extremal problem $\min_{a_j:\,a_1=1} \left( -iF(i) \right) = \min_{a_j:\,a_1=1} \sum\limits_{j=1}^{N} {(-1)^{j+1} a_j}.$ We show that the solution is $\frac12 \sec^2{\fracπ{2N+2}},$ and the extremal polynomial $\sum_{j = 1}^N \frac{U'_{2(N-j+1)} \left( \cos\left(\fracπ{2N+2}\right)\right)}{U'_{2N} \left( \cos\left(\fracπ{2N+2}\right)\right)}z^{2j-1}$ is unique and univalent, where the $U_j(x)$ are the Chebyshev polynomials of the second kind and $U'_j(x)$ denotes the derivative. As an application, we obtain the estimate of the Koebe radius for the odd univalent polynomials in $\mathbb D$ and formulate several conjectures.

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