Extremal problems for typically real odd polynomials

On the class of typically real odd polynomials of degree \(2N-1\) $2N-1$ \(F(z)=z+\sum_{j=2}^Na_jz^{2j-1}\) $$F(z)=z+\sum _{j=2}^N{a}_j{z}^{2j-1}$$ we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient \(a_2.\) $a_2.$ It is shown that \(\begin{gathered} |{F(z)}|\le \frac12\csc^2\left({\frac{\pi}{2N+2}}\right),\\-1+4\sin^2\left({\frac{\pi}{2N+4}}\right)\le a_2\le-1+4\cos^2\left({\frac{\pi}{N+2}}\right) \quad \text{for odd $N$,}\end{gathered} \) $$\begin{array}{c}|F(z)|\le \frac{1}{2}{csc}^2\left(\frac{\pi }{2N+2}\right),\\ -1+4{sin}^2\left(\frac{\pi }{2N+4}\right)\le a_2\le -1+4{cos}^2\left(\frac{\pi }{N+2}\right)\text{for odd}N\text{,}\end{array}$$ and \(-1+4(\nu_N)^2\le a_2\le -1+4\cos^2\left({\frac{\pi}{N+2}}\right) \quad \text{for even $N$,}\) $$-1+4{({\nu }_N)}^2\le a_2\le -1+4{cos}^2\left(\frac{\pi }{N+2}\right)\text{for even}N\text{,}$$ where \(\nu_N\) ${\nu }_N$ is a minimal positive root of the equation \(U'_{N+1}(x) = 0\) $U_{N+1}^{\prime }(x)=0$ with \(U'_{N + 1}(x)\) $U_{N+1}^{\prime }(x)$ being the derivative of the Chebyshev polynomial of the second kind of the corresponding order.The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined.