Problem Statement
Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is contained in $\{z: \lvert f(z)\rvert< 1\}$.
Determine the behaviour of $\rho(f)$. In particular, is it always true that $\rho(f)\gg 1/n$?
Determine the behaviour of $\rho(f)$. In particular, is it always true that $\rho(f)\gg 1/n$?
Categories:
Analysis Polynomials
Progress
A problem of Erdős, Herzog, and Piranian, who note that $f(z)=z^n-1$ has $\rho(f) \leq \frac{\pi/2}{n}$.Pommerenke [Po61] proved that\[\rho(f) \geq \frac{1}{2en^2}.\]Krishnapur, Lundberg, and Ramachandran [KLR25] proved\[\rho(f) \gg \frac{1}{n\sqrt{\log n}}.\]
Source: erdosproblems.com/1039 | Last verified: January 19, 2026