  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.
