How to prove this inequality $$\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$$ for $f>0$.
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2$\begingroup$ Riemann sums and AM-GM. $\endgroup$– anonDec 30, 2013 at 21:25
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3$\begingroup$ Convexity. ${}{}$ $\endgroup$– Pedro ♦Dec 30, 2013 at 21:56
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2 Answers
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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.
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You can find the proof in https://www.math.ucdavis.edu/~hunter/m218b_09/Lp_and_Sobolev_notes.pdf on page 3.
