Erdős Problem #1046

Let \(f\in \mathbb{C}[x]\) be a monic polynomial and \[E=\{ z: \lvert f(z)\rvert <1\}.\] If \(E\) is connected then is \(E\) contained in a disc of radius \(2\)? A problem of Erdős, Herzog, and Piranian [EHP58], who also ask, if \(\{ z: \lvert f(z)\rvert\leq 1\}\) is connected, then what are the least possible diameter and greatest possible width of this set, and conjecture the answer is \(2\) in both cases. Their guess that the width is always at most \(2\) is false, as Pommerenke [Po59] gave an example with width \(>\sqrt{3}2^{1/3}\approx 2.18\). The condition that \(E\) is connected is equivalent to \(E\) containing all zeros of \(f'\). The answer is yes, and in fact the centre of this disc can be taken to be \(\frac{z_1+\cdots+z_n}{n}\), where the \(z_i\) are the roots of \(f\), as shown by Pommerenke [Po59].