I have $n$ non-negative integers $x_1, \dotsc, x_n$ which satisfy the constraint $\sum x_i = S$
I want to derive a bound on $\sum x_i^2$. An easy bound can be calculated as:
$\sum x_i^2 \le (max_{x_i}) \sum x_i = S^2$
This bound works for non-negative reals. Is there a tighter bound for non-negative integers or is this the best we can do?
If the integers are not necessarily distinct, then you can create an upper bound by making all but one of the integers $0$, and make the last integer $S$, which gives you an upper bound of $S^2$. This is the best we can do, as increasing any of the zero integers will get us a lower result by the fact that $(S-n)^2+n^2-S^2=S^2-2Sn+2n^2-S^2=2n(n-S)$, which is negative except at $n=0,S$, at which points the difference between the result and the upper bound is zero.
