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 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).

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

Related posts Volume of Lp norm balls Swedish superellipse Squircles [*] Note that r is only required to be greater than 0; analysis books typically focus on r ≥ 1.


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