lebesgue integral problem 2

liyi

$f:[0,1]\to[1,\infty)$ is a measurable function.
Show that
$\left(\int_{[0,1]} f(x)dx\right)\left(\int_{[0,1]} \ln f(x)dx\right) \leq \left(\int_{[0,1]} f(x) \ln f(x)dx\right)$

**Posted:** Sun Dec 26, 2004 5:59 pm

dickchimney · Poincare Conjecture Offline Joined: 06 Mar 2004 Posts: 218

**Posted:** Mon Dec 27, 2004 3:25 am

Kent Merryfield

Yes. dickchimney's proof is completely correct, and shows that the original stipulation on the range of $f$ could be weakened to $f:[0,1]\to(0,\infty),$ as long as all of the integrals in dickchimney's proof converge absolutely.

**Posted:** Mon Dec 27, 2004 8:35 am

liyi

[quote="dickchimney]
$\left(\int_{[0,1]} f(x)dx\right)\ln \left(\int_{[0,1]}f(x)dx\right) \geq \left(\int_{[0,1]} f(x)dx\right)\left(\int_{[0,1]} \...$
[/quote]
How to obtain this?

**Posted:** Tue Dec 28, 2004 6:15 pm

grobber

I think he did mention that it follows from the concavity of $\ln$ .

**Posted:** Tue Dec 28, 2004 6:23 pm

liyi

thanks.
i realized it soon after posting it....
//shamed

**Posted:** Tue Dec 28, 2004 6:25 pm