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\[\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.\]

#225: [Er40b,p.957][Er57][Er61,p.248][Ha74]

analysis

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 [SSS73] (for general $c_k\in \mathbb{C}$).

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This page was last edited 29 December 2025.

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Additional thanks to: Winston Heap, Vjekoslav Kovac, and Karlo Lelas

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