Sum of positive elements divided by their "weighted" product - inequality 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.
 A: Concavity of $\log$ gives
$$\log\left( \prod_{i=1}^nx_i^{p_i}\right) = \sum_{i=1}^n p_i\log x_i \stackrel{\mbox{concavity}}{\leq} \log\left( \sum_{i=1}^np_i x_i\right) \stackrel{0\leq p_i\leq 1}{\leq} \log\left( \sum_{i=1}^nx_i\right)$$
A: Applying the weighted AM-GM inequality to the weights $p_1, p_2, \ldots, p_n$, we obtain
\begin{equation}
\dfrac{p_1 x_1 + p_2 x_2 + \cdots + p_n x_n}{1} \geq \sqrt[1]{x_1^{p_1} x_2^{p_2} \cdots x_n^{p_n}}
\end{equation}
(since $p_1 + p_2 + \cdots + p_n = \sum_{i=1}^n p_i = 1$). This simplifies to
\begin{equation}
p_1 x_1 + p_2 x_2 + \cdots + p_n x_n \geq x_1^{p_1} x_2^{p_2} \cdots x_n^{p_n} .
\end{equation}
Hence,
\begin{align}
x_1^{p_1} x_2^{p_2} \cdots x_n^{p_n}
\leq p_1 x_1 + p_2 x_2 + \cdots + p_n x_n
= \sum_{i=1}^n \underbrace{p_i}_{\substack{\leq 1 \\ \text{(since $p_i \in \left[0,1\right]$)}}} x_i \leq \sum_{i=1}^n x_i ,
\end{align}
so that
\begin{align}
\sum_{i=1}^n x_i \leq x_1^{p_1} x_2^{p_2} \cdots x_n^{p_n} = \prod_{i=1}^n x_i^{p_i} .
\end{align}
