# Stolarsky Approximations to Popular Means

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The Stolarsky mean provides approximations to popular means, which shows that it generalizes some of them and that these means are strictly ordered by strict inequality over on .

Contributed by: Robert L. Brown (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The Stolarsky mean is is derived from the mean value theorem and its form depends upon the independent parameter . As the parameter is varied, approximations can be made to popular means. Most approximations become exact for some value of ; the Stolarsky is a true generalization for those means. Strict ordering of the Pythagorean Means (Wolfram *MathWorld*) is the basis of many proofs in number theory.

Note that, although we consider means of two positive numbers, we only need to consider for , because each of these means is a homogeneous function: where .

Like the Hölder mean, the Stolarsky mean parameter will approximate, match exactly, or match in the limit many popular means. The following table contrasts these two generalized forms.

Bookmarks provide the most uncluttered way to examine the estimation error for a given mean. Additional reference can be obtained by clicking a mean's name next to its checkbox.

## Permanent Citation