If $a_1,\dots,a_n$ are all unequal positive quantities, then prove that:
$$\prod_{i=1}^n a_i^{a_i} > \left(\frac{\sum_{i=1}^n a_i}{n}\right)^{\sum_{i=1}^n a_i}$$
No other conditions are given.
I tried to solve it using logarithms, but I could not understand how can I prove $${a_1\log a_1}+{a_2 \log a_2}+\dots+{a_n\log a_n} > {(a_1+\dots +a_n)} \left(\log(a+\dots+a_n)-\log(n)\right)$$
And please tell me, is there any standard method to solve this kind of problems?