Erdős Problem #225

Let \[ f(\theta) = \sum_{0\leq k\leq n}c_k e^{ik\theta}\] be a trigonometric polynomial all of whose roots are real, such that \(\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1\). Then \[\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.\] This is Problem 4.20 in [Ha74], where it is attributed to Erdős. This was solved independently by Kristiansen [Kr74] (only in the case when \(c_k\in\mathbb{R}\)) and Saff and Sheil-Small [SaSh74] (for general \(c_k\in \mathbb{C}\)). (The original proof of Kristiansen contained an error which was later fixed in [Kr76].) "All roots real" refers to the zeros of \(f\) as an entire function of \(\theta\in\mathbb{C}\). We normalise by \(c_0c_n\neq 0\) and \(n\geq 1\): a factor \(e^{im\theta}\) affects neither the zeros nor \(\lvert f\rvert\) on the real line, while a single term \(c_me^{im\theta}\) has no zeros and \(\int_0^{2\pi}\lvert f\rvert = 2\pi\).