I have a short question.
Is it possible to compute the following:
$ \inf \left\lbrace \frac{\sqrt{\sum_{i=1}^n x_i}}{\sum_{i=1}^n \sqrt{x_i}} \, \middle| \, \forall i: x_i>0 \right\rbrace $
An upper bound for this value is $ 1/\sqrt{n} $, but I'm not sure whether this is also the value of the infimum.
By the root-mean square inequality (a special case of the generalized AM-GM):
$$ \sqrt{\frac{\sum_{i=1}^n x_i}{n}} \;\ge\; \frac{\sum_{i=1}^n \sqrt{x_i}}{n} \quad \iff \quad \frac{\sqrt{\sum_{i=1}^n x_i}}{\sum_{i=1}^n \sqrt{x_i}} \ge \frac{1}{\sqrt{n}} $$
Equality is attained iff $\;x_i = x_j\;$ so $\;\cfrac{1}{\sqrt{n}}\;$ is not only an infimum, but an actual minimum as well.