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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). A polynomial (1) is $P(z)=\prod_{j=1}^{n}(1-z/\omega_j)$ with every $\omega_j$ on the unit circle $C$, and the inequalities (2) are

$$\lvert P(z)\rvert\le\bigl\lvert1-\lvert z\rvert^n\bigr\rvert \qquad\text{and}\qquad \lvert P(z)\rvert\ge1+\lvert z\rvert^n .$$

The problem (p. 347). Determine the greatest degree $n$ for which every polynomial (1) of degree $n$ satisfies the first inequality of (2) on one and the second on another of two appropriate radii of the unit disc, or of two half-lines from the origin. The paper states both readings, radii and half-lines, without choosing between them.

What the paper proves (p. 349). By Theorem 2, for $n\le4$ two such half-lines always exist. Theorem 2 is stated for the monic form $\prod(z-z_r)$, which differs from (1) by a unimodular constant factor and so has the same modulus.

Read depth. Claims checked: the paragraph of p. 347 was read clause by clause on the page image of the print. Nothing here is independently reviewed.

Proof pointer

The paper proves the cases $n\le4$ (Theorem 2, pp. 349--351) and leaves the problem open beyond them.

Dependencies

Theorem 2.

Bears on

The paper links this problem to no Erdős problem in the corpus.