We prove that $$\begin{aligned} \frac{5}{6} + \sum_{k=1}^{n} \frac{\cos k \theta}{k} \ge \frac{1}{4} (1+\cos \theta )^2 \quad (n=2,3,\ldots;\ \theta \in (0, \pi )), \end{aligned}$$ 5 6 + ∑ k = 1 n cos k θ k ≥ 1 4 ( 1 + cos θ ) 2 ( n = 2 , 3 , … ; θ ∈ ( 0 , π ) ) , where equality holds if and only if $$n = 2$$ n = 2 and $$\theta = \pi - \cos^{-1} \frac{1}{3}$$ θ = π - cos - 1 1 3 . This refines a result of Brown and Koumandos.

中文翻译：

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杨氏余弦多项式的函数界限

我们证明 $$\begin{aligned} \frac{5}{6} + \sum_{k=1}^{n} \frac{\cos k \theta}{k} \ge \frac{1}{ 4} (1+\cos \theta )^2 \quad (n=2,3,\ldots;\ \theta \in (0, \pi )), \end{aligned}$$ 5 6 + ∑ k = 1 n cos k θ k ≥ 1 4 ( 1 + cos θ ) 2 ( n = 2 , 3 , … ; θ ∈ ( 0 , π ) ) ，其中当且仅当 $$n = 2$$ n = 时等式成立2 和 $$\theta = \pi - \cos^{-1} \frac{1}{3}$$ θ = π - cos - 1 1 3 。这改进了 Brown 和 Koumandos 的结果。