An extremal problem for odd univalent polynomials

Abstract: 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 $\frac12 \sec^2{\frac{\pi}{2N+2}},$ and the extremal polynomial [ \sum_{j = 1}^N \frac{U'{2(N-j+1)} \left( \cos\left(\frac{\pi}{2N+2}\right)\right)}{U'{2N} \left( \cos\left(\frac{\pi}{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.