How to prove this inequality $$\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$$ for $f>0$.
2026-03-29 10:08:54.1774778934
Integral inequality $\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$
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$\log$ is concave. this is just Jensen's inequality. See http://en.wikipedia.org/wiki/Jensen's_inequality Look at the measure theoretic form. Please check that this makes sense to you.