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,

x = (x_1, x_2, \ldots, x_n)

and let r > 0 [*]. Then the r-mean is defined to be

M_r(x) = \left( \frac{1}{n} \sum_{i=1}^n x_i^r \right)^{1/r}

and the r-sum is define to be

 S_r(x) = \left( \sum_{i=1}^n x_i^r \right)^{1/r}

These definitions come from the classic book Inequalities by Hardy, Littlewood, and Pólya, 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, 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).

Plot of M_r and S_r

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

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