Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have the following function $f:[0,\frac{1}{2}] \to \mathbb{R}$:

$$f(p) = p^2(\log(p))^2 - (1-p)^2(\log(1-p))^2 + (1-2p)\log(p)\log(1-p) + (1-2p)\{p\log(p)+(1-p)\log(1-p)\}$$

The inequality I need to show is $$f(p) \leq 0$$I can show that $f(0) = f(1/2) = 0$, and that $f'(0) = -1$, $f'(1/2) = 0$. The graph of $f$ looks like

valid xhtml.

Since its not monotonic/convex/concave I'm stuck. Any leads are welcome!

share|cite|improve this question
The domain of $f$ doesn't include $0$ because of the $\log(p)$ terms. How did you get $f(0)=0$? –  Patrick Apr 11 '12 at 0:16
In the limit as p goes to 0.. I should have been more clear though. –  VSJ Apr 11 '12 at 0:57
Can you prove that $f'$ has only one zero? –  lhf Apr 11 '12 at 2:30
@lhf: I tried that, but the derivative did not look manageable enough. –  VSJ Apr 11 '12 at 2:46
Using laws of logs, you can simplify the equation. You might be then able to take the derivative, as @lhf suggested. –  Ben Apr 19 '12 at 8:55

1 Answer 1

up vote 1 down vote accepted

I managed to solve this eventually in a not so elegant way, for the proof outline and more details about where this inequality came from please refer the mathoverflow link

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.