Integral inequality
Posted: Sat Dec 10, 2016 10:42 am
Let $f:[0, 1] \rightarrow \mathbb{R}$ be a differentiable function with decreasing derivative satisfying $f(0)=0$ and $f'(1)>0$. Prove that:
$$\int_0^1 \frac{1}{1+f^2(t)}\, {\rm d}t \leq \frac{f(1)}{f'(1)}$$
$$\int_0^1 \frac{1}{1+f^2(t)}\, {\rm d}t \leq \frac{f(1)}{f'(1)}$$