## Problem of the week

### 2018-21 AM-GM inequality

Does there exist a (possibly $$n$$-dependent) constant $$C$$ such that $\frac{C}{a_n} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \leq \frac{a_1+ \dots + a_n}{n} - \sqrt[n]{a_1 \dots a_n} \leq \frac{C}{a_1} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2$ for any $$0 < a_1 \leq a_2 \leq \dots \leq a_n$$?