**Update:** A year ago, but the first answer is not clear with me. I bounty this question again.

My question:I am looking for a proof or counterexample to the following inequality:

If $n \in \mathbb{N}$, and $a_1 \ge a_2 \ge \cdots \ge a_n \ge 0$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 0$ then

$${\left(\sum_{i=1}^{n}{a_i^{\alpha_i}} \right)}^n \ge \prod_{i=1}^{n}{\left(\sum_{j=1}^{n}{a_i^{\alpha_j}} \right)}$$

If $n \in \mathbb{N}$ and $a_1 \ge a_2 \ge \cdots \ge a_n \ge 0$ and $0 \le \alpha_1 \le \alpha_2 \le \cdots \le \alpha_n$ then

$${\left(\sum_{i=1}^{n}{a_i^{\alpha_i}} \right)}^n \le \prod_{i=1}^{n}{\left(\sum_{j=1}^{n}{a_i^{\alpha_j}} \right)}$$

Moreover equality holds in either case if and only if $a_1=a_2=a_3=...=a_n$.

[The original question claimed a proof when $n=2$ or $n=3$.]