Extending the result $\left(\frac{a+b}{2}\right)^{a+b} \lt a^a b^b$ to $n$ number of terms.
How to prove the inequality $a_1^{a_1}a_2^{a_2}\cdots a_n^{a_n} > \left(\frac{a_1+\dots+a_n}{n}\right)^{a_1+\dots+a_n}$?