Hölder Approximations to Popular Means
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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 .
Contributed by: Robert L. Brown (March 2011)
Open content licensed under CC BY-NC-SA
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 .
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