Power Mean Inequality
Revision as of 12:51, 17 June 2006 by PenguinIntegral (talk | contribs) (First edit, full of errors :))
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 ,
- $M(k) = \left( \frac{1}{n} \sum_{i=1}^n a_{i}^k \right) ^ {\frac{1}{k}}
/math>
(The case k=0 is taken to be the geometic mean)
=== Inequality ===
If −∞ ≤ ''a'' < ''b'' ≤ ∞, then M(''a'') ≤ M(''b''). Equality if and only if ''a''<sub>1</sub> = ''a''<sub>2</sub> = ... = ''a''<sub>''n''</sub>, following from <math>\frac{\partial M(t)}{\partial t}\geq 0$ (Error compiling LaTeX. Unknown error_msg) for −∞ ≤ t ≤ ∞, proved with Jensen's inequality.
