What is the better inequality (below and above) for the next expression:
$$\sum_{(i_1, \ldots , i_N) \in S_N ^{(n)}} \prod_{j=1} ^ N i_j \quad?$$
where $S_N ^{(n)}$ is the set of permutations of $N$ elements take from $n.$
I try using the first and last $N$ terms of $[n]$ respectively, but it is not enough because I lose a lot. And I don't know if the AM–GM inequality work due to the equality between AM–GM $(\text{when}\quad i_1=i_2=i_=\ldots)$ knowing that in this case it's not possible for being permutation.
Thanks in advance.
P.S.: Sorry for my English, I'm not native. :v