It would seem that sums and means are trivially related; the mean is just the sum divided by the number of items. But when you generalize things a bit, means and sums act differently.
Let x be a list of n non-negative numbers,
and let r > 0 [*]. Then the r-mean is defined to be
and the r-sum is define to be
Here’s the theorem alluded to in the title of this post:
As r increases, Mr(x) increases and Sr(x) decreases.
If x has at least two non-zero components then Mr(x) is a strictly increasing function of r and Sr(x) is a strictly decreasing function of r. Otherwise Mr(x) and Sr(x) are constant.
The theorem holds under more general definitions of M and S, such letting the sums be infinite and inserting weights. And indeed much of Hardy, Littlewood, and Pólya is devoted to studying variations on M and S in fine detail.
Here are log-log plots of Mr(x) and Sr(x) for x = (1, 2).
Note that both curves asymptotically approach max(x), M from below and S from above.
[*] Note that r is only required to be greater than 0; analysis books typically focus on r ≥ 1.