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Let $r_k>0$ for $k = 1,\ldots, n$, let $\alpha_k, \beta_k\in \mathbb{R}$ be given such that $|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that $\left|\sum\limits_{k=1}^nr_ke^{i(\alpha_k+\epsilon_k\beta_k)}\right|\le 1$ for all choices $\epsilon_k=\pm1$. How to prove $$\sum\limits_{k=1}^nr_k\le n\sin\left(\frac{\pi}{2n}\right)?$$

The inequality is known, but the proof is rather complicated. So I am looking for a concise proof.

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You say it's known. Can you say where the proof is to be found? –  Anthony Quas Dec 7 '12 at 3:58
    
Here it is. Proposition 8 in Linear Algebra and its Applications 428 (2008) 305–315. –  Betrand Dec 7 '12 at 14:38
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