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Kristiansen proves Erdős's conjecture in its two-sided form (Proc. Amer. Math. Soc. 44 (1974), 49--57, statement on p. 49). For a trigonometric polynomial $T$ of degree $n\ge1$ with real coefficients and $2n$ real roots, the mean of $|T|$ between two consecutive roots is at most $\frac2\pi\max|T|$, so

$$\int_{-\pi}^{\pi}|T(x)|\,dx\le4\max|T|.$$

This gives the one-sided display of [[problems/analysis/E0225/_index|Problem 225]] for arbitrary complex coefficients, an observation recorded by this corpus. If $P(z)=\sum_{k=0}^nc_kz^k$, $n\ge1$, $c_n\ne0$, has its $n$ roots $e^{i\alpha_j}$ on the unit circle, then $|f(\theta)|=|c_n|2^n|T(\theta/2)|$ with $T(\varphi)=\prod_j\sin(\varphi-\alpha_j/2)$. This $T$ is a real trigonometric polynomial of degree $n$ with $2n$ real roots, and $\int_0^{2\pi}|f|=|c_n|2^n\int_{-\pi}^{\pi}|T|\le4\max|f|$. The site's description, the case $c_k\in\mathbb R$, is narrower than the paper's statement. The general complex case is also proved, by a different route, by Saff and Sheil-Small on their page.

Correction. The site states that Kristiansen's original proof contained an error later fixed in [Kr76]. The erratum (Proc. Amer. Math. Soc. 58 (1976), 377) says that the proof of the companion note Proof of a polynomial conjecture (Proc. Amer. Math. Soc. 44 (1974), 58--60) is incomplete. That note concerns real polynomials with all roots in an interval, and its proof misses one case, which the erratum says can be treated by methods similar to those of the trigonometric paper. The erratum reports no gap in the trigonometric paper, which cites only Erdős's 1940 note. The site's commentary thus attaches the companion note's erratum to this paper.

Acceptance. The paper is refereed: G. K. Kristiansen, Proof of an inequality for trigonometric polynomials, Proc. Amer. Math. Soc. 44, no. 1 (May 1974), 49--57. The site's curator, Thomas F. Bloom, marks Problem 225 proved and credits this paper with an independent solution of the case $c_k\in\mathbb R$ only, which is narrower than the paper's theorem, so the curator's credit is not listed as reviewed. The page is dated by the first day of the issue month, since the paper's first posting carries no finer date.