Let $f\in \mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a constant small enough such that $\{ z: \lvert f(z)\rvert\leq c\}$ has $m$ distinct connected components.

Let $f\in \mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a constant small enough such that $\{ z: \lvert f(z)\rvert\leq c\}$ has $m$ distinct connected components.

Must all these components be convex?

A question of Grunsky, which was reported by Erdős, Herzog, and Piranian [EHP58].

The answer is no, as shown by Pommerenke [Po61], who showed that, if $k$ is sufficiently large, and $f(z)=z^k(z-a)$ where $a$ is sufficiently close to $(1+\frac{1}{k})k^{\frac{1}{k+1}}$, then $\{ z: \lvert f(z)\rvert\leq 1\}$ has two components, and the component which contains $0$ is not convex.