SOLVED
This has been resolved in some other way than a proof or disproof.
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 $\Lambda(f)$.
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 $n=2$).
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This page was last edited 16 January 2026.
Additional thanks to: Quanyu Tang
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #1044, https://www.erdosproblems.com/1044, accessed 2026-01-16
