An extremal problem for odd univalent polynomials

For the univalent polynomials $F(z)={\sum }_{j=1}^N{a}_j{z}^{2j-1}$ with real coefficients and normalization $a_1=1$ we solve the extremal problem $$\underset{a_j:a_1=1}{\min }(-iF(i))=\underset{a_j:a_1=1}{\min }\sum _{j=1}^N(-1{)}^{j+1}{a}_j\mathrm{.}$$ We show that the solution is $\frac{1}{2}{\sec }^2\frac{\pi }{2N+2},$ and the extremal polynomial $$\sum _{j=1}^N\frac{U_{2(N-j+1)}^{\mathrm{\prime }}(\cos \left(\frac{\pi }{2N+2}\right))}{U_{2N}^{\mathrm{\prime }}(\cos \left(\frac{\pi }{2N+2}\right))}{z}^{2j-1}$$ is unique and univalent, where the $U_j(x)$ are the Chebyshev polynomials of the second kind and $U_j^{\mathrm{\prime }}(x)$ 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.