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Studying for a convex optimization exam I encountered the below question. I suspect the inequality can be proved using Jensen inequality with the function $f(x)=-\ln(1-x)$ and $x_i=a_i$ but can't work it out.

True or false?

Let $n>1$. For any $a_1, ..., a_n>0$ such that $\sum_{i=1}^n a_i=1$, the following inequality holds:

$$\prod_{i=1}^n [a_i(1-a_i)] \leq \left(\frac{n-1}{n^2}\right)^{\!n}$$

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Hint: $x\mapsto \log x(1-x)$ is concave. Now use Jensen.

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  • $\begingroup$ @Dole The given conditions ensure $a_i \in (0,1)$. $\endgroup$ – Macavity Feb 14 at 18:27
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Just for reference, without Jensen, by AM-GM:

$\prod_i a_i(1-a_i) \le \frac{(\sum_i a_i(1-a_i))^n}{n^n}=\frac{(1-\sum_i a_i^2))^n}{n^n}$ [1]

Plugging in the inequality beweenn the 2-mean and arithmetic mean:

$\sum_i a_i^2 \ge \frac{1}{n}$ [2]

Combining [2] and [1] we do the same job as Jensen.

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