# Inequality involving $\log$

Let $g$ be a non-negative measurable function on $[0,1]$. How can I show that $$\int \log ~(g(u))~\text{d}u \leq \log~\int g(u)~\text{d}u$$ whenever the left hand side is defined.

If it helps, I know that the $\log$ function is concave.

Edit:
Can I argue like this: Since $\log$ concave, $-\log$ is convex; So by Jensen's inequality, we have $$-\log~\int g(u)~\text{d}u ~\leq~ \int-\log ~(g(u))~\text{d}u?$$

• Presumably, you are considering a version of $\log$ that takes values in the extended real line? Nov 10 '11 at 16:52
• Yes, Jensen's inequality can be used, in the opposite direction, with concave functions, just as you've shown in your Edit.
– robjohn
Nov 10 '11 at 17:46
• A much nicer problem is to show that $\lim_{p \to 0} \|f\|_{L^p} = \exp \left ( \int_X \log |f(x)| \, \textrm{d}\mu(x) \right )$. Nov 10 '11 at 18:06