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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; $|E|$ is area.

Problem 2 (pp. 133--134). "To determine the polynomials (1) whose zeros lie in $\bar D$ and which have the property that $|E|$ takes on its least value $\alpha_n$ for the fixed degree $n$. Also, to obtain an estimate of $\alpha_n$; for example, does there exist a positive constant $c$ such that $\alpha_n>n^{-c}$?"

The paper adds (p. 134) that the opposite problem is solved: for any zeros, $|E|\le\pi$, with the supremum attained only when all zeros coincide (Pólya, its [6], p. 280); and it recalls Cartan's theorem (its [1], p. 273) that $E$ can be covered by at most $n$ disks whose radii sum to less than $2e$.

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 2 on pp. 133--134. The copy read is identified on the source card.

Read depth. Claims checked: the problem and the remarks after it were read clause by clause on the page images of pp. 133--134 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

The Corollary to Theorem 4, which shows that $\alpha_n$ is not bounded below by a positive constant (the step is drawn on that page).

Bears on

• #116: the problem's question whether the area exceeds $n^{-O(1)}$ for zeros with $|z_i|\le1$ is the second question of Problem 2. The alternative bound $(\log n)^{-O(1)}$ in the problem's statement does not appear in the paper.