Maximal Polynomial Ranges for a Domain of Intersection of Two Circular Disks

Abstract

Let Ω be a domain in the complex plane $\mathbb{C}$ containing the origin and let ${\cal P}_{n}^{0}(\Omega)$ denote the set of all polynomials P of degree ≤ n satisfying the conditions $P(0)=0$ and $P(\mathbb{D})\subset \Omega$, where $\mathbb{D}=\left\{z\,:\,\,|z|<1\right\}$ is the unit disk in the complex plane. The maximal range, denoted by Ω _n , is then defined as the union of all sets $P(\mathbb{D}), P \in {\cal P}_{n}^{0}$ .We shall construct extremal polynomials and maximal polynomial range for a domain of intersection of two circular disks.