On a problem of Erdős, Herzog and Piranian (Q1813165)

The author deals with the lemiscate domain \(S(f)\); \(\{| f(z)|\leq 1\}\). \textit{P. Erdős}, \textit{F. Herzog} and \textit{G. Piranian} [J. Anal. Math. 6, 125--148 (1958; Zbl 0088.25302)] studied metric properties of \(S(f)\) and made several conjectures. The author establishes estimates for the linear measure of the intersection of \(S(f)\) with the unit circle \(C\). In the paper cited above it is conjectured that this linear Lebesgue measure \(m(S(f)\cap C)\) can be made arbitrarily small for a special choice of polynomial \(f(z)\). An additional conjecture [Erdős et al. (loc. cit.), pp. 132 and 134) deals with the lower bound. Both conjectures are proved by the author for polynomials with all the zeros on the unit circle. He establishes theorem 1. For each \(f(z)\) with all the zeros on the unit circle \(m(S(f)\cap C)>{1\over 4\sqrt n}\). Also there exists such an \(f(z)\) that \(m(S(f)\cap C)< 16(\log n/n)^{1/3}\). Similar results are obtained for polynomials with all their zeros in \([-2, 2]\).