Difference between revisions of "Power Mean Inequality"
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M(k) = \left( \frac{1}{n} \sum_{i=1}^n a_{i}^k \right) ^ {\frac{1}{k}} | M(k) = \left( \frac{1}{n} \sum_{i=1}^n a_{i}^k \right) ^ {\frac{1}{k}} | ||
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(The case k=0 is taken to be the geometic mean) | (The case k=0 is taken to be the geometic mean) | ||
Revision as of 13:01, 17 June 2006
The Mean
The power mean inequality is a generalized form of the multi-variable AM-GM inequality.
The kth "Power Mean", with exponent k and a series (a_i) of positive real numbers is ,
(The case k=0 is taken to be the geometic mean)
=== Inequality ===
If −∞ ≤ a < b ≤ ∞, then M(a) ≤ M(b). Equality if and only if a1 = a2 = ... = an, following from 
for −∞ ≤ t ≤ ∞, proved with Jensen's inequality.
