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Source. Section 1, p. 347, of P. Erdős, F. Herzog and G. Piranian, Polynomials whose zeros lie on the unit circle, Duke Math. J. 22 (1955), 347--351, DOI 10.1215/S0012-7094-55-02237-7, the edition named on the source card.

Statement

Setting (p. 347). $C$ is the unit circle and a polynomial (1) is $P(z)=\prod_{j=1}^{n}(1-z/\omega_j)$ with every $\omega_j$ on $C$. Cohen's theorem, as the paper states it, gives for every such $P$ a path from the origin to $C$ on which "the inequality $\lvert P\rvert<1$ holds everywhere except at $z=0$" (p. 347).

The question (p. 347, quoted). "Does there exist a universal constant $L$ such that for every polynomial (1) the inequality $\lvert P\rvert<1$ holds on a path which connects the origin to $C$ and has length at most $L$?"

Answer reported (p. 347). The paper says the question was recently answered in the negative by G. R. MacLane, citing his paper On a conjecture of Erdős, Herzog, and Piranian, Michigan Math. J. 2 (1953--1954), 147--148 (reference [2]). The paper raises the question in connection with Theorem 1.

Read depth. Claims checked: the paragraph of p. 347 and reference [2] on p. 351 were read clause by clause on the page images of the print. MacLane's paper was not read here.

Proof pointer

The paper proves nothing about the question; it reports MacLane's answer and cites it.

Dependencies

Cohen, Modulus of an analytic function, Amer. Math. Monthly 59 (1952), 704--705 (reference [1]), for the path from the origin to $C$; MacLane (reference [2]) for the answer.

Bears on

• Problem 1215: the problem asks this question, with $C$ for the paper's $L$ and the polynomials normalized by $P(0)=1$ with all roots on the unit circle. The paper poses it and reports MacLane's negative answer without proving it.