**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 $n=3$.]



A rearrangement inequality for exponentiation function
The first inequality is not true for $n=2$, for example
as =



```
0.0912    0.2256

```

alphas =



```
0.8281    1.7010

```

LHS =



```
0.0471

```

RHS =



```
0.0574

```

found by the following Matlab/Octave program:


n=2;


for iter=1:1e4
as=cumsum(rand(1,n));
alphas=cumsum(rand(1,n));


LHS=sum(as.^alphas)^n;
RHS=prod(sum(as.^(alphas')));


if LHS

as
 alphas
 LHS
 RHS


end


end




