Estimating the Koebe radius for polynomials

TL;DR

This paper determines the extremal coefficients of conjugate trigonometric polynomials to estimate the Koebe radius, providing explicit solutions and implications for polynomial image coverage of the unit disc.

Contribution

It introduces a unique solution for the extremal problem involving conjugate trigonometric polynomials and connects it to the Koebe radius estimation, with explicit formulas using Chebyshev polynomials.

Findings

Explicit extremal coefficients derived using Chebyshev polynomials

The supremum of the minimal real part is -1/4 sec^2(pi/(N+2))

Results imply new theorems on interval coverage by polynomial images

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