Estimating the Koebe radius for polynomials

Abstract

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 \[ \sup_ {a_2,...,a_N} \left ({ \min_t \left\{ {\Re \left ({ F\left ({ { e^ {it} } } \right) } \right): \Im \left ({ F\left ({ { e^ {it} } } \right) } \right) = 0 } \right\} } \right) = -\frac14 \sec ^2\fracπ{N + 2}. \] We show that the solution is unique and is given by \[ a_j^ {(0)} = \frac {1} { { { U'_N}\left ({\cos \frac{π} { { N + 2 } } } \right) } } { U' _ { N - j + 1 } }\left ({\cos \frac{π} { { N + 2 } } } \right) { U_ { j - 1 } }\left ({\cos \frac{π} { { N + 2 } } } \right), \] where the $U_j(x)$ are the Chebyshev polynomials of the second kind, and the $U_j^{\prime }(x)$ are their derivatives, $j=1,\dots ,N.$ 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.