The site's display admits degenerate cases for which the bound fails. The constant $f\equiv1$ ($n=0$) has no roots, maximum $1$ and integral $2\pi$. If roots are read as zeros of $f$ as an entire function of $\theta$, $f(\theta)=e^{im\theta}$ fails the same way. The problem's sources state the conjecture with every root counted. Kristiansen states it for a trigonometric polynomial of degree $n\ge1$ with real coefficients and $2n$ real roots (Proc. Amer. Math. Soc. 44 (1974), p. 49). Saff and Sheil-Small state it for one of degree $n$ with all $2n$ zeros in $[0,2\pi)$ real (their Conjecture 1). The corrected Statement follows them: $n\geq1$ and $c_n\neq0$, so that $n$ is the actual degree of $P(z)=\sum_{k=0}^n c_kz^k$, and every one of the $n$ algebraic roots of $P$, counting multiplicity, on the unit circle, of the form $e^{i\theta}$ with $\theta$ real. The site's display does not state a root count; the full-root reading is the hypothesis used in the exact transfer below. The corrected Statement is proved, by Kristiansen [Kr74] (a two-sided real theorem, which the site describes as the real-coefficient case) and by Saff and Sheil-Small [SaSh74] in general (see Status). The site's wording fails at the degenerate cases above, and no other result about it is recorded.

Saff and Sheil-Small also prove an equivalent two-sided theorem. Their degree-$n$ trigonometric polynomial has $2n$ real zeros in a period and is converted to a degree-$2n$ algebraic polynomial. That normalization is recorded separately rather than identified with the one-sided display.