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Statement

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

Problem 12 (p. 142). "For fixed degree $n$ of $f$, is the length of the lemniscate $|f(z)|=1$ greatest in the case where $f(z)=z^n-1$? Is the length at least $2\pi$, if $E$ is connected? What is the infimum of the length for polynomials (1) whose zeros all lie in $D$?"

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 12 on p. 142. 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. 142 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

None.

Bears on

• #114: the problem is the first question of Problem 12, in the same terms. The second and third questions are not part of it.