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Claim. For a monic polynomial $f$ whose open set $E=\{|f(z)|<1\}$ is connected, the closed set $\{|f(z)|\le1\}$ of Problem 509 is covered by one disc of radius $2$, so the answer is yes for every such $f$. Theorem 3 of Pommerenke's 1959 note, as printed on p. 222 with $C$ the lemniscate $|f(z)|=1$: "Let $\zeta=(z_1+\cdots+z_n)/n$, where $z_1,\cdots,z_n$ are the zeros of $f(z)$. If $E$ is connected, then $C$ is contained in the circle $|z-\zeta|<2$." Since $E$ is bounded and its boundary lies on $C$, the set $\{|f|\le1\}=E\cup C$ lies in the same open disc (an elementary remark, not the paper's sentence). The proof (pp. 222--223) applies a Pólya--Szegő bound to the inverse of $f^{1/n}$, which is univalent outside the unit disc because $E$ is connected. The statement is on the result page theorem_3 of the source card pommerenke_1959_some_problems_erdos_herzog_piranian; the same theorem is the full answer to Problem 1046, which asks whether a connected $E$ lies in some disc of radius $2$; the theorem answers it with the disc about the centroid of the zeros.

Covers. Every monic $f$ whose open set $\{|f|<1\}$ is connected. The site's remark states the hypothesis for the set itself, the closed set $\{|f|\le1\}$, a weaker hypothesis that Theorem 3 does not state (the closed set is connected whenever the open set is, but not conversely); the closed case is proved in Pommerenke's 1961 paper, on Pommerenke 1961. The general question, every monic $f$, is untouched; the general bounds of Cartan ($2e$) and Pommerenke ($2.59$, [Po60] on the problem page) settle no instance of it.

Source. Chr. Pommerenke, On some problems by Erdös, Herzog and Piranian, Michigan Math. J. 6 (1959), no. 3, 221--225, DOI 10.1307/mmj/1028998227; received January 15, 1959. The publisher's record dates the article to the year 1959 alone, and the page is named by the record's date.

Acceptance. Refereed: the note appeared in the Michigan Mathematical Journal. The site's remarks credit the paper 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. Nothing here is independently reviewed by this project.

Depends on. No page of this wiki. The proof rests on the cited paper and on a Pólya--Szegő problem (Aufgaben und Lehrsätze, Vol. 2, Section IV, Problem 140), which is not held.