Hi, if I understand well your result can be written like this:

r = (sum mi * si) / (sum mi) / (sum si)

Thus you have a dot product divided by the 1-norms of vectors m and s (I suppose mi > 0 and si > 0).

Thanks to Cauchy Swartz inequality the absolute value of the dot product is bounded by the product of the 2-norms of vectors m and s.

In addition we have for any vector x of dimension n:

2-norm(x)