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Statement

$f(z)=\prod_{\nu=1}^n(z-z_\nu)$ and $E=\{|f(z)|\le1\}$ (p. 97).

Theorem 10 (pp. 106--107). "Let $z_0=\frac1n\sum_{\nu=1}^nz_\nu=0$ and $\sigma^2=\frac1n\sum_{\nu=1}^n|z_\nu|^2$. Then the following best possible results hold:

(a) If $|z_\nu|\le\sqrt2/2$ or if $z_\nu\in[-1,+1]$, then $E$ is connected.

(b) If $E$ is connected, then $|z_\nu|<2$ and $\sigma<\sqrt2$."

Inside the proof of (b) the paper states the containment the problem pages use (p. 107): with $z_0=0$ and $E$ connected, "Hence $E$ is contained in $|z|\le2$ (see for instance [6, p. 42]), and it follows that $|z_\nu|<2$ because $z_\nu$ is an interior point of $E$." Since a translation of the zeros translates $E$, a connected lemniscate set lies in the closed disk of radius 2 about the centroid of the zeros. This containment answers Problem 14 of the 1958 paper (whether the lemniscate set of a $K$-polynomial lies in a disk of radius 2 centered at the centroid) affirmatively; the paper does not name Problem 14 here, and the author had already answered it as Theorem 3 (p. 222) of his 1959 note [9], pommerenke_1959_some_problems_erdos_herzog_piranian.

Sharpness (pp. 107--108): $(z^2-1/2)^m(z^2+a^2)$ with $a>\sqrt2/2$ has three components for large $m$, and $z^2-a^2$ with $a>1$ has two; the Chebyshev polynomial $T_n(2^{1/n-1}z)=z^n+\cdots$ has a connected $E$ with a zero $2^{1-1/n}\cos(\pi/2n)\to2$ and $\sigma^2=2^{2-2/n}\cdot\frac1n\sum\cos^2\frac\pi{2n}(2\nu-1)\to2$.

Source. Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97--115; Theorem 10 on printed pp. 106--107 (PDF pp. 10--11 of the publisher's scan), its proof on pp. 107--108 (PDF pp. 11--12), Lemma 4 on pp. 105--106 (PDF pp. 9--10), read on the page images (the scan has no text layer). The copy read is identified in the source digest.

Read depth. Claims checked: the statement, the containment sentence and the sharpness examples were read clause by clause on the page images. The proof of (a) (a paragraph) was followed; the proof of (b) and Lemma 4 were read for structure and not checked. Nothing here is independently reviewed.

Proof pointer

(a) (p. 107): for $|z_\nu|\le\sqrt2/2$, Lemma 1 (pp. 98--99) with $z_0=0$ and $\sigma^2\le1/2$ puts the disk $|z|\le\sqrt2/2$ in $E$, and Lemma 2 (p. 99) makes $E$ connected; for $z_\nu\in[-1,1]$, both halves $[-1,0]$ and $[0,1]$ lie in $E$ by [2, Theorem 1], and Lemma 2 applies again.

(b) (p. 107): with $z_0=0$, $w=f(z)^{1/n}=z+a_2^*z^{-1}+\cdots$ (6) is univalent in the exterior $\{|f|>1\}$ of the connected $E$, so its inverse $z=\phi(w)=w+\sum_{\mu\ge1}b_\mu w^{-\mu}$ (7) is meromorphic and univalent in $|w|>1$; the Koebe-type bound for such functions gives $E\subset\{|z|\le2\}$ (Golusin [6, p. 42]) and $|z_\nu|<2$. From (5), the integral of Lemma 4 evaluates to $\lambda(r)/n=r^2+\sum|b_\mu|^2r^{-2\mu}$ for $r>1$, so Lemma 4 gives $\sigma^2<\lambda(1)/n=1+\sum|b_\mu|^2$, and the area theorem $\sum\mu|b_\mu|^2\le1$ (Golusin [6, p. 39]) gives $\sigma^2<2$.

Dependencies

Within the paper: Lemmas 1, 2 and 4. Outside it: Theorem 1 of the 1958 paper for the interval case (erdos_1958_metric_properties_polynomials), and for (b) the distortion bound and the area theorem for univalent functions in $|w|>1$ from Golusin, Geometrische Funktionentheorie (1957), pp. 42 and 39 (not held).

Bears on

• Problem 509: the connected case. When $\{|f|\le1\}$ is connected it lies in one disk of radius 2 centered at the centroid of the zeros (p. 107), so a single circle of radius 2 covers it; the paper states nothing about the disconnected case, which is the problem's open content. The card of Hong 2026 recalls this connected case.
• Problem 1038: context only. For real zeros in $[-1,1]$ with centroid $0$, (a) records that $[-1,1]\subset E$ by [2, Theorem 1]; the paper adds no bound on the measure of $E\cap\mathbb R$ for that class.