Inequality with a sum of constants

Real Analysis
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Riemann
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Location: Melbourne, Australia

Inequality with a sum of constants

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Post by Riemann »

Let $x \in \left( - \frac{\pi}{2} , \frac{\pi}{2} \right)$ and consider the function

\[f(x) = a_1 \tan x + a_2 \tan \frac{x}{2} + \cdots +a_n \tan \frac{x}{n}\]

where $a_1, a_2, \dots, a_n \in \mathbb{R}$ and $n \in \mathbb{N}$. If $\left| f(x) \right| \leq \left| \tan x \right|$ for all $x \in \left( - \frac{\pi}{2} , \frac{\pi}{2} \right)$ then prove that

\[\left| a_1 + \frac{a_2}{2} + \cdots + \frac{a_n}{n} \right| \leq 1\]
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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