I'm wondering, where the following inequality is arising from, that I read in a book: $$\left\lvert \frac{\lambda_{1}+\lambda_{1}\lambda_{2}+...+\lambda_{q-2}\lambda_{q-1}+\lambda_{q-1}\lambda_{q}}{1+\lambda_{1}^2+\lambda_{2}^2+...+\lambda_{q-1}^2+\lambda_{q}^2} \right\rvert \leq \cos(\frac{\pi}{q+2})$$ I have no clue, where the cosinus is coming from. Note that all the $\lambda$'s $\in \mathbb{R}$. For small $q$ it is easily possible to show that by differentiating the function on the lhs and finding the maximum. But for $q$ bigger than 2 this gets really difficult. Can anyone help me understanding this?
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Let $\vec{a} = (1, \lambda_1, \dotsc, \lambda_{q-1})$ and $\vec{b} = (\lambda_1, \dotsc, \lambda_q)$. I think the inequality is related to $\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|,|\vec{b}|}$. – André Caldas Jun 02 '22 at 10:20
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Look at this: https://math.stackexchange.com/questions/807326/how-to-prove-this-inequality-fraca-1a-2a-2a-3-cdotsa-n-1a-na – Mateo Jun 02 '22 at 10:58