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Claim. Let $p(z)=\prod_{i=1}^n(z-z_i)$ with every $\lvert z_i\rvert\le1$. Then $\{z:\lvert p(z)\rvert\le1\}$ contains a disk of radius $(2e)^{-1}n^{-2}$ about one of the zeros, so

$$\lvert\{z:\lvert p(z)\rvert<1\}\rvert\ge\pi\,(2e)^{-2}\,n^{-4},$$

the open set included, since every point of that disk has modulus below $1$. This is Theorem 4 of Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97–115 (Theorem 4, printed p. 101), recorded on the result page [[../library/analysis/pommerenke_1961_metric_properties_complex_polynomials/theorem_4|Theorem 4]] of the card [[../library/analysis/pommerenke_1961_metric_properties_complex_polynomials/_index|Pommerenke 1961]]. It answers the question of Problem 116 yes: the area is at least $n^{-O(1)}$, with exponent $4$. The proof combines the diameter bound of the paper's Theorem 3 (the component through $0$ has diameter above $1$, hence capacity above $1/4$) with the derivative bound $\lvert p'\rvert<2en^2$ on that component, and integrates from a zero to the nearest point of modulus $1$. The parenthetical stronger form, an area of at least $(\log n)^{-O(1)}$, is not covered by this claim; it is the subject of the accepted claim [[problems/polynomials/E0116/claims/2025_03_24_krishnapur_lundberg_ramachandran|Krishnapur, Lundberg and Ramachandran 2025]].

Depends on. No page of this wiki: the proof is self-contained in the paper.

Acceptance. Refereed: the paper appeared in the Michigan Mathematical Journal in 1961 (volume 8, issue 2; the issue carries no month, so the page's date is the first day of the publication year). Reviewed: erdosproblems.com labels the problem proved and states that the lower bound $\gg n^{-4}$ follows from this paper, cited as [Po61] (page last edited 2025-10-24), which the corpus counts as documented independent acceptance by the site's curator, T. F. Bloom (erdosproblems.com); the community database lists the problem as proved with a Lean marker (entry last updated 2026-08-24).

Formalization. The site's Lean marker refers to a formal proof by others, with a different argument: the file linked above, in the lean-proofs repository at the pinned commit, ends with Erdos116.erdos_116, which states that there are $c>0$ and $C\in\mathbb N$ with $\lvert\{z:\lvert\prod_i(z-a_i)\rvert<1\}\rvert>c\,n^{-C}$ for every $n>0$ and every $a$ with all $\lvert a_i\rvert\le1$, obtained with the explicit constants $c=\pi/2^{31}$ and $C=14$; its docstring describes an elementary reflected-polynomial and finite-Fourier argument rather than Pommerenke's, and its header names Codex and GPT-5.6 Sol as formal authors and Christian Pommerenke, this paper's author, as the informal author, which is why the file is a link on this page and not a claim of its own. The formal-conjectures statement of the problem is marked solved with a link to this declaration. The development is unbuilt and unaudited against the question by this corpus, so the evidence lists no formalized kind; the standing rests on the refereeing and the site's acceptance.