Extremal problems for typically real odd polynomials

On the class of typically real odd polynomials of degree \(2N-1\)

$$F(z)=z+\sum_{j=2}^Na_jz^{2j-1}$$

we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient \(a_2.\) It is shown that

$$\begin{gathered} |{F(z)}|\le \frac12\csc^2\left({\frac{\pi}{2N+2}}\right),\\-1+4\sin^2\left({\frac{\pi}{2N+4}}\right)\le a_2\le-1+4\cos^2\left({\frac{\pi}{N+2}}\right) \quad \text{for odd $N$,}\end{gathered} $$

and

$$-1+4(\nu_N)^2\le a_2\le -1+4\cos^2\left({\frac{\pi}{N+2}}\right) \quad \text{for even $N$,}$$

where \(\nu_N\) is a minimal positive root of the equation \(U'_{N+1}(x) = 0\) with \(U'_{N + 1}(x)\) 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.