Problem 1041

Erdős Problem #1041 [status: falsifiable; formalized: yes]. Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert < 1$ for all $i$. Must there always exist a path of length less than $2$ in$$\{z: \lvert f(z)\rvert < 1\}$$which connects two of the roots of $f$? Prize: no. Tags: analysis.