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 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, *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 Pólya 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.

## Related posts

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