Problem Statement
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 $\Lambda(f)$.
Categories:
Analysis
Progress
A problem of Erdős, Herzog, and Piranian [EHP58].This has been resolved by Tang, who proved that the infimum of $\Lambda(f)$ over all such $f$ is $2$. Tang also suggests that, if the degree $n$ is fixed, then the the infimum over all such $f$ of degree $n$ is attained by $f_n(z)=z^n-1$ (and proves this for $n=1$ and $n=2$).
Source: erdosproblems.com/1044 | Last verified: January 19, 2026