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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!

  • 0
    Using laws of logs, you can sim$p$lify the equ$a$tion. You might be then able to take the derivative, as @lhf suggested.2012-04-19

1 Answers 1

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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 https://mathoverflow.net/questions/93271/proving-a-messy-inequality