Chebyshev polynomials on equipotential curves

Abstract

For an analytic function $\varphi (z)$ with a Laurent expansion at $\infty$ of the form \begin{equation*} ϕ(z)=z+c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots, \end{equation*} the Faber polynomial $F_n$ of degree $n$ associated to $\varphi$ is the polynomial part of the Laurent series at $\infty$ of $\varphi (z{)}^n$. We prove that the $n$th Chebyshev polynomial $T_{n,L_r}$ for the equipotential curve $L_r=\{z\in \mathbb{C}:|\varphi (z)|=r\}$ converges to $F_n$ as $r\to \infty$. The proof makes use of the fact that zero is the strongly unique best approximation to the monomial $z^n$ on the unit circle by polynomials of degree less than $n$.