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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
    The domain of $f$ doesn't include $0$ because of the $\log(p)$ terms. How did you get $f(0)=0$?2012-04-11
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    In the limit as p goes to 0.. I should have been more clear though.2012-04-11
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    Can you prove that $f'$ has only one zero?2012-04-11
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    @lhf: I tried that, but the derivative did not look manageable enough.2012-04-11
  • 0
    Using laws of logs, you can simplify the equation. You might be then able to take the derivative, as @lhf suggested.2012-04-19

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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