Problem 1044

Erdős Problem #1044: declared status 'solved'. Formalized: no. Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$. If $\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of$$\{ z: \lvert f(z)\rvert<1\}$$then determine the infimum of $\Lambda(f)$. Prize: no. Tags: analysis.