Problem 1120

Erdős Problem #1120: declared status 'open'. Formalized: no. Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$. Let$$E= \{ z : \lvert f(z)\rvert \leq 1\}.$$What is the shortest length of a path in $E$ joining $z=0$ to $\lvert z\rvert =1$? Current best: The trivial lower bound for the length of this path is $1$, which is achieved for $f(z)=z^n$. Prize: no. Tags: analysis.