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Statement

Notation (p. 125): $f$ is a monic polynomial (1), $E$ the set where $|f|<1$, $\bar D$ the closed unit disk.

Problem 3 (p. 134). "For the polynomials (1) with all $z_\nu$ in $\bar D$, let $\rho_n$ denote the radius of the largest disk which is necessarily contained in $E$. What is the asymptotic behavior of $\rho_n$? Does there exist a positive constant $c$ such that $\rho_n>c/n$? The example $f(z)=z^n-1$ shows that the constant $c$ can not be greater than $\pi/2$."

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 3 on p. 134. The copy read is identified on the source card.

Read depth. Claims checked: the problem was read clause by clause on the page image of p. 134 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

None. Theorem 6 gives a disk of fixed radius when the zeros lie in a closed set of transfinite diameter below $1$, which excludes $\bar D$.

Bears on

• #1039: the problem's questions are those of Problem 3, stated for $\rho(f)$ of a single polynomial; the paper's $\rho_n$ is the radius guaranteed for every $f$ of degree $n$, and its bound $c\le\pi/2$ from $z^n-1$ is part of the source.