An extremal problem for odd univalent polynomials

TL;DR

This paper solves an extremal problem for odd univalent polynomials with real coefficients, finding the maximum of a specific linear functional, and applies the result to estimate the Koebe radius.

Contribution

It provides an explicit solution and proof of extremality for a class of odd univalent polynomials, including the unique extremal polynomial form.

Findings

The extremal value is  \, ext{sec}^2{rac{\u03c0}{2N+2}}.

The extremal polynomial is expressed via derivatives of Chebyshev polynomials.

Application to estimating the Koebe radius for odd univalent polynomials.

Abstract

For the univalent polynomials $F(z) = \sum\limits_{j=1}^{N} a_j z^{2j-1}$ with real coefficients and normalization \(a_1 = 1\) we solve the extremal problem \[ \min_{a_j:\,a_1=1} \left( -iF(i) \right) = \min_{a_j:\,a_1=1} \sum\limits_{j=1}^{N} {(-1)^{j+1} a_j}. \] We show that the solution is $\frac12 \sec^2{\frac{\pi}{2N+2}},$ and the extremal polynomial \[ \sum_{j = 1}^N \frac{U'_{2(N-j+1)} \left( \cos\left(\frac{\pi}{2N+2}\right)\right)}{U'_{2N} \left( \cos\left(\frac{\pi}{2N+2}\right)\right)}z^{2j-1} \] is unique and univalent, where the $U_j(x)$ are the Chebyshev polynomials of the second kind and $U'_j(x)$ denotes the derivative. As an application, we obtain the estimate of the Koebe radius for the odd univalent polynomials in $\mathbb D$ and formulate several conjectures.