Difference between revisions of "Power Mean Inequality"
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| − | + | === Inequality === | |
If −∞ ≤ ''a'' < ''b'' ≤ ∞, then M(''a'') ≤ M(''b''). Equality if and only if ''a''<sub>1</sub> = ''a''<sub>2</sub> = ... = ''a''<sub>''n''</sub>, following from <math>\frac{\partial M(t)}{\partial t}\geq 0</math> for −∞ ≤ ''t'' ≤ ∞, proved with [[Jensen's inequality]]. | If −∞ ≤ ''a'' < ''b'' ≤ ∞, then M(''a'') ≤ M(''b''). Equality if and only if ''a''<sub>1</sub> = ''a''<sub>2</sub> = ... = ''a''<sub>''n''</sub>, following from <math>\frac{\partial M(t)}{\partial t}\geq 0</math> for −∞ ≤ ''t'' ≤ ∞, proved with [[Jensen's inequality]]. | ||
Revision as of 13:13, 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.
