Estimating the Koebe radius for polynomials

For a pair of conjugate trigonometrical polynomials $C (t) = \sum_ { j = 1 } ^N { { a_j}\cos jt }, S(t) = \sum_ { j = 1 } ^N { { a_j}\sin jt }$ with real coefficients and normalization $a_1=1$ we solve the extremal problem $$\underset{a_2,\mathrm{.}\mathrm{.}\mathrm{.},a_N}{\sup }\left(\underset{t}{\min }\left\{\mathrm{\Re }\left(F\left(e^{it}\right)\right):\mathrm{\Im }\left(F\left(e^{it}\right)\right)=0\right\}\right)=-\frac{1}{4}{\sec }^2\frac{\pi }{N+2}\mathrm{.}$$ We show that the solution is unique and is given by $$a_j^{(0)}=\frac{1}{U_N^{\mathrm{\prime }}\left(\cos \frac{\pi }{N+2}\right)}{U}_{N-j+1}^{\mathrm{\prime }}\left(\cos \frac{\pi }{N+2}\right)U_{j-1}\left(\cos \frac{\pi }{N+2}\right),$$ where the $U_j(x)$ are the Chebyshev polynomials of the second kind, and the $U_j^{\mathrm{\prime }}(x)$ are their derivatives, $j = 1, \ldots, N.$ As a consequence, we obtain some theorems on covering of intervals by polynomial images of the unit disc. We formulate several conjectures on a number of extremal problems on classes of polynomials.