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Claim. For a monic polynomial $f$ whose closed set $E=\{|f(z)|\le1\}$ is connected, $E$ lies in the closed disc of radius $2$ about the centroid of the zeros, so one disc of radius $2$ covers it and the answer to Problem 509 is yes for every such $f$. Pommerenke's 1961 paper writes $E$ for the closed set (p. 97). Theorem 10(b) (pp. 106--107) assumes the centroid of the zeros at $0$ and concludes, for a connected $E$, that every zero has modulus below $2$; its proof (p. 107) first states that $E$ is contained in $|z|\le2$, citing Golusin's distortion bound for functions univalent outside the unit disc, applied to the inverse of $f^{1/n}$. A translation of the zeros translates $E$, so the containment holds about the centroid in general. The statement and the containment are on the result page theorem_10 of the source card pommerenke_1961_metric_properties_complex_polynomials.

Covers. Every monic $f$ whose closed set $\{|f|\le1\}$ is connected. This is the case the site's remark names, and it contains the class of Pommerenke 1959, since the closed set is connected whenever the open set $\{|f|<1\}$ is. The general question, every monic $f$, is untouched.

Depends on. No page of this wiki. The proof uses Golusin's distortion bound for functions univalent outside the unit disc, which the paper cites and which is not held.

Acceptance. Refereed: Michigan Math. J. 8 (1961), no. 2, 97--115. The site's remarks credit Pommerenke with the connected case, but the site labels the problem OPEN, so the curator's label settles neither the problem nor a part of it, and no reviewed evidence is listed.