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 $\int_0^1 \frac{dx}{1+e^{2x}} = 0.28 < 1.$