Is anything known about the behavior of the function $$f(x)=\prod\_{i=1}^n x\_i^{x\_i}$$ on the standard simplex, i.e. the set $\{x\in\mathbb{R}^n:\sum\_{i=1}^n x\_i=1, x\_i\geq0\}$? I ask because I have done a lot of numerical experiments that suggest that $$1\leq\left(\prod\_{i=1}^{n}x\_{i}^{x\_{i}}\right)\left(\sum\_{i=1}^{n}x\_{i}^{1/2}\right)^{2}\leq 2$$ for all $n$ and all $x$ on the standard simplex, and I have no clue how to prove this. Of course, one could also re-write this more compactly as $$1\leq\left(\prod\_{i=1}^{n}x\_{i}^{x\_{i}}\right)\|x\|\_{1/2}\leq 2$$but this still doesn't seem particularly helpful to me.



An inequality on the simplex involving $x^x$
The upper bound is not correct. E.g., let $n=101$, $x\_1=\dots=x\_{100}=1/1000$, $x\_{101}=9/10$. Then 
$$\left(\prod\_{i=1}^{n}x\_{i}^{x\_{i}}\right)\left(\sum\_{i=1}^{n}x\_{i}^{1/2}\right)^{2}>7>2.$$


More generally, take any $a\in(0,1)$ and any natural $k$, and then let $n=k+1$, $x\_1=\dots=x\_k=a/k$, $x\_{k+1}=1-a$. Then
$$\ln\left[\left(\prod\_{i=1}^{n}x\_{i}^{x\_{i}}\right)\left(\sum\_{i=1}^{n}x\_{i}^{1/2}\right)^{2}\right]=a\ln\frac ak+(1-a)\ln(1-a)+2\ln(\sqrt{ak}+\sqrt{1-a})\to\infty$$
as $k\to\infty$. 


