Sum and mean inequalities move in opposite directions

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

These definitions come from the classic book Inequalities by Hardy, Littlewood, and Polya, except the authors use the Fraktur forms of M and S. If r = 1 we have the elementary mean and sum.

Here's the theorem alluded to in the title of this post:

As r increases, M_r(x) increases and S_r(x) decreases.

If x has at least two non-zero components then M_r(x) is a strictly increasing function of r and S_r(x) is a strictly decreasing function of r. Otherwise M_r(x) and S_r(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 Polya is devoted to studying variations on M and S in fine detail.

Here are log-log plots of M_r(x) and S_r(x) for x = (1, 2).

Note that both curves asymptotically approach max(x), M from below and S from above.

• Volume of L^p norm balls
• Swedish superellipse
• Squircles

[*] Note that r is only required to be greater than 0; analysis books typically focus on r >= 1.