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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 5 (p. 139). "If all the $z_\nu$ lie in $D$, does there exist a path of length less than 2 which lies in $E$ and joins two of the $z_\nu$?"

Just before it the paper notes (p. 139) that the remarks opening Section 5 (p. 136, where Szegő's bound of $n-1$ components of $\bar E$ is cited) show that when all zeros lie in $D$ some component of $E$ contains at least two zeros.

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 5 on p. 139. The copy read is identified on the source card.

Read depth. Claims checked: the problem and the sentence before it were read on the page image of p. 139 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

None.

Bears on

• #1041: the problem is Problem 5, with the zeros in the open disk $|z_i|<1$ as in the paper.