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
Explore with Wolfram|Alpha
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 Power Mean
Cite this as:
Cantrell, David W. and Weisstein, Eric W. "Power Mean." From MathWorld --A Wolfram
Web Resource. https://mathworld.wolfram.com/PowerMean.html
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