I have following expression,
$$ \frac{\sum_{i=1}^n x_i}{\prod_{i=1}^nx_i^{p_i}} $$
where $p_i$s satisfy $\sum p_i = 1$ and $p_i \in [0,1]$ and $x_i\geq0$, $\forall i \in 1\dots n$.
I think that this expression is always $\geq 1$, however, I don't know how to prove it.
Is there anything I can conclude?
Thanks.