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I wonder what is the functions family that satisfies the following inequality:

$\int_0^1 \frac{dx}{1+f^{2}(x)} <\frac{f(1)}{f'{(1)}}$

This inequality seems to be a very interesting inequality, but not sure when it works and when not. For example, it works if i take $f(x)=e^x$ that is $\int_0^1 \frac{dx}{1+e^{2x}} = 0.28 < 1.$

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    Ok, I see. Thanks!2012-07-09

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If we assume that: $f'$ is continuous and not increasing; $f'(1) \neq 0$; $f(0) = 0$; $f(1) \ge 0$, than: $\int_0^1 \frac{f'(x)}{1 + f^2 (x)} \cdot \frac{dx}{f'(x)} \le \frac{\arctan f(1)}{f'(1)} \le \frac{f(1)}{f'(1)}$