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

The Hölder mean is often called the generalized mean or power mean. As the power is varied, approximations can be made to popular means. Most approximations become exact for some value of ; the Hölder is a true generalization for those means. Strict ordering of the Pythagorean means is the basis of many proofs in number theory.

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

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