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Source. The unnumbered Theorem, p. 23 (also stated in the abstract, p. 23), proof pp. 24--25, of P. Erdős and E. Netanyahu, A remark on polynomials and the transfinite diameter, Israel J. Math. 14 (1973), 23--25, DOI 10.1007/BF02761531, the edition named on the source card.

Statement

Setting (p. 23). For complex numbers $z_1,\dots,z_n$ put $f(z)=\prod_{\nu=1}^{n}(z-z_\nu)$, and let $E(f)$ be the set of $z$ with $|f(z)|<1$ (the paper's (1) and (2)).

Theorem (p. 23, quoted). "Let $D$ be a bounded, closed and connected set, whose transfinite diameter $d(D)$ is equal to $1-c$, $0<c<1$. Let $E(f)$ be the point set defined by (2), with $z_\nu\in D$, $\nu=1,\cdots,n$. Then there exists a positive number $\rho=\rho(c)$ (dependent only on $c$) such that the set $E(f)$ always contains a disk of radius $\rho(c)$."

So one radius serves every degree $n$ and every choice of zeros in $D$; the radius depends on $D$ only through $c$. The paper says a weaker result is Theorem 6 of Erdős, Herzog and Piranian (J. Analyse Math. 6 (1958), 125--148), that its proof is an existence proof, and that a numerical estimate for $\rho$ would be interesting (p. 23).

Remarks of the paper (p. 25).

• The theorem is false without connectedness; the paper points to the lemniscate $|z^2-a^2|<1$, $a>0$, where increasing $a$ makes the radius of every disc in $E(z^2-a^2)$ as small as one pleases.
• The theorem implies that for $D$ connected of transfinite diameter $1-c$ and all $z_\nu\in D$ the area of the closure of $E(f)$, the set where $|f(z)|\le1$, is greater than some $f(c)$; the paper has no explicit estimate of $f(c)$.
• For $D$ of transfinite diameter $1$ the paper suggests ("perhaps") that the area of the closure of $E(f)$ can be made smaller than any $\varepsilon>0$ once $n>n_0(\varepsilon)$, connectedness then not being needed; it records that Erdős, Herzog and Piranian proved this when $D$ is the unit circle or the interval $(-2,+2)$, and says the general case is open.
• It also raises the maximum number of components of the closure of $E(f)$: for $D$ the unit circle it recalls from Erdős, Herzog and Piranian (their Theorem 7) that the maximum is $n-1$, for $D$ the interval $(-2,+2)$ it says without proof that $E(f)$ can have $n$ components, and it says the general case had not, as far as the authors knew, been investigated.

Read depth. Claims checked: the statement, its setting and the remarks were read clause by clause on the page images of pp. 23 and 25. The proof was read in outline only and not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pages 24--25, written here in outline. Take the polynomial $P(z)=\prod_{i=1}^m(z-t_i)$ of degree $m=m(c)$ given by the Lemma, with $|P|<\tfrac12$ on $D$. By continuity there is $\rho>0$ such that moving each $t_i$ to any point $s_i$ of the disc $H_i$ of radius $\rho$ about $t_i$ keeps $\prod_i|z-s_i|$ below $\tfrac12+\varepsilon$ on $D$. Choose $s_i$ maximizing $|f|$ on $H_i$. Up to sign, $\prod_i f(s_i)$ is the product over the zeros $z_\nu$ of $\prod_i(z_\nu-s_i)$, which has modulus less than $1$, so some $|f(s_i)|$ is at most $1$, and $|f|<1$ throughout that $H_i$. The paper says the argument follows Theorem 6 of Erdős, Herzog and Piranian.

Dependencies

The Lemma (pp. 23--24) of the same paper.

Bears on

• Problem 1040: the problem asks whether $\mu(F)$, the infimum of the area of $\{z:|f(z)|<1\}$ over monic $f$ with all zeros in a closed infinite set $F$, is determined by the transfinite diameter of $F$, and whether $\mu(F)=0$ when that diameter is at least $1$. For bounded, closed, connected $D$ of transfinite diameter $1-c$ with $0<c<1$ the theorem puts a disc of radius $\rho(c)$ inside every such set $\{z:|f(z)|<1\}$, and the paper notes the resulting lower bound, depending only on $c$, for the area of the closed set $\{z:|f(z)|\le1\}$. For transfinite diameter $1$ it only suggests that this area can be made arbitrarily small for large $n$, records that Erdős, Herzog and Piranian proved this for the unit circle and the interval $(-2,+2)$, and calls the general case open. The paper does not use the problem's numbering.
• Problem 1042: the p. 25 remark on the number of components of the closed set $\{z:|f(z)|\le1\}$ records the unit-circle maximum $n-1$ from Erdős, Herzog and Piranian and states without proof that for zeros in the interval $(-2,+2)$ the set $E(f)$ can have $n$ components. It proves nothing on the problem.