Is it possible for a number $\alpha>1$ to have such a kind of inequality? $$\int_A f^{-\alpha} \leq C \left( \int_A f\right)^{-\alpha},$$ where the measure of $A$ is finite $A$ and $f\geq 0$ is a positive function.
If not, is there any kind of inequality one could exploit for $\int_A f^{-\alpha}$ in roder to get rid of the exponent $-\alpha$ inside the integral?
Thanks a lot!
I don' think that such an estimate exists. As an example, choose $f(x) = 2x$ on $A = (0, 1)$. Then the right-hand side is $C$ whereas the left-hand side is $\infty$ for every $\alpha > 1$.