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Statement

$f(z)=\prod_{\nu=1}^n(z-z_\nu)$, $E=\{|f(z)|\le1\}$, and $\rho$ is "the radius of the largest disk contained in $E$" (p. 101). Quoted (p. 101): "If $\operatorname{cap}A<1$, there exists a positive number $\rho_0=\rho_0(A)$ such that $\rho\ge\rho_0$ [2, Theorem 6]. If $A$ is a disk of radius 1 or a segment of length 4 (in both cases, $\operatorname{cap}A=1$), there does not exist any positive lower bound for $\rho$ that is independent of the degree $n$ of $f(z)$. Erdös, Herzog and Piranian put the question whether $\rho\ge\mathrm{const}\cdot n^{-1}$ if $|z_\nu|\le1$ [2, Problem 3]. I shall only prove a weaker estimate (see also [2, Problem 2])."

Theorem 4 (p. 101). "If $|z_\nu|\le1$, the lemniscate domain $E$ contains a disk of radius $(2e)^{-1}n^{-2}$."

The disk found is centered at a zero $z_\mu$ and lies in the component $E_0$ of $E$ containing $0$. Its open interior lies in the interior of $E$, which is the open set $\{|f|<1\}$, so that set has area at least $\pi(2e)^{-2}n^{-4}$; this consequence is drawn here, not printed.

Source. Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97--115; Theorem 4 with its proof and the introductory paragraph on printed p. 101 (PDF p. 5 of the publisher's scan), read on the page image (the scan has no text layer). The copy read is identified in the source digest.

Read depth. Claims checked: the statement and the introductory paragraph were read clause by clause on the page image; the proof (one paragraph) was read in full and its steps followed, with the external inputs named below taken as cited. Nothing here is independently reviewed.

Proof pointer

Page 101. Let $E_0$ be the component of $E$ containing $0$. Theorem 3 with $r=1$ gives $d_0>1$ for its diameter; since $E_0$ is connected, $d_0\le4\operatorname{cap}E_0$ (Golusin [6, p. 42]), so $\operatorname{cap}E_0>1/4$. Since $|f|\le1$ on $E_0$, the author's derivative bound [11] gives $|f'(z)|\le en^2/(2\operatorname{cap}E_0)<2en^2$ for $z\in E_0$. Take a zero $z_\mu\in E_0$ and the boundary point $z^*$ of $E_0$ nearest to it; integrating $f'$ along the segment, $1=|f(z^*)|<|z^*-z_\mu|\cdot2en^2$, so $|z^*-z_\mu|>1/(2en^2)$ and the disk $|z-z_\mu|\le1/(2en^2)$ lies in $E_0\subset E$.

Dependencies

Within the paper: Theorem 3 at $r=1$ (theorem_3). Outside it: the inequality $\operatorname{diam}\le4\operatorname{cap}$ for a continuum (Golusin, Geometrische Funktionentheorie, 1957, p. 42; not held) and the derivative bound $|f'|\le en^2/(2\operatorname{cap}E_0)$ on a component where $|f|\le1$, from the author's On the derivative of a polynomial, Michigan Math. J. 6 (1959), 373--375 (cited as [Po59a] on Problem 115's page; not held).

Bears on

• Problem 116: the area of $\{|f|<1\}$ is at least $\pi(2e)^{-2}n^{-4}>n^{-O(1)}$, the polynomial lower bound the problem asks for; the paper says nothing about the $(\log n)^{-O(1)}$ strengthening, later obtained by Krishnapur, Lundberg and Ramachandran 2025, whose card names this theorem as the bound it improves.
• Problem 1039: $\rho(f)\ge(2e)^{-1}n^{-2}$, the "weaker estimate" the paper proves toward the $\rho\gg1/n$ of [2, Problem 3], which is the problem's question and which the paper leaves open.