A power mean is a mean of the form
 |
(1) |
where the parameter 
is an affinely extended real number
and all 
.
A power mean is also known as a generalized mean, Hölder mean, mean of degree
(or order or power) 
,
or power mean.
The following table summarizes some common named means that are special cases of the generalized mean, where
 |
(2) |
and
The plots above visualize the generalized mean by plotting the special values
 |
(7) |
with 
red, 
orange, 0 black, 1 green, 2 blue, and 
violet.
See also
Arithmetic Mean, Geometric Mean, Harmonic Mean, Mean, Pythagorean Means, Root-Mean-Square
Portions of this entry contributed by David W. Cantrell
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References
Borwein, J. M. and Borwein, P. B. "General Means and Iterations." Ch. 8 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Bullen, P. S. "The Power Means." Ch. 3 in Handbook of Means and Their Inequalities. Dordrecht, Netherlands: Kluwer, 2003.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities. Cambridge, England: Cambridge University Press, 1952.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 121, 2003.
Referenced on Wolfram|Alpha
Cite this as:
Cantrell, David W. and Weisstein, Eric W. "Power Mean." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PowerMean.html