Integral inequality

Real Analysis
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Riemann
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Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

Integral inequality

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Let $f:[0, 1] \rightarrow \mathbb{R}$ be a differentiable function with decreasing derivative satisfying $f(0)=0$ and $f'(1)>0$. Prove that:

$$\int_0^1 \frac{1}{1+f^2(t)}\, {\rm d}t \leq \frac{f(1)}{f'(1)}$$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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