# Proving inequality using Jensen

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:

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

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

• @Dole The given conditions ensure $a_i \in (0,1)$. – Macavity Feb 14 at 18:27

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.