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.

  • $\begingroup$ @Dole The given conditions ensure $a_i \in (0,1)$. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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