Power Mean
المؤلف:
Borwein, J. M. and Borwein, P. B.
المصدر:
"General Means and Iterations." Ch. 8 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.
الجزء والصفحة:
...
30-6-2019
1627
Power Mean
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
 |
symbol |
mean |
 |
 |
minimum |
 |
 |
harmonic mean |
 |
 |
geometric mean |
 |
 |
arithmetic mean |
 |
RMS |
root-mean-square |
 |
 |
maximum |
The plots above visualize the generalized mean by plotting the special values
![M_p(x,1)=<span style=]() {x for p=-infty; (2x)/(1+x) for p=-1; sqrt(x) for p=0; (1+x)/2 for p=1; sqrt((1+x^2)/2) for p=2; 1 for p=infty " src="http://mathworld.wolfram.com/images/equations/PowerMean/NumberedEquation3.gif" style="height:212px; width:222px" /> |
(7)
|
with 
red, 
orange, 0 black, 1 green, 2 blue, and 
violet.
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.
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