Difference between revisions of "Power Mean Inequality"
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Revision as of 14:11, 26 October 2007
The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.
Inequality
For a real number 
and positive real numbers 
, the th power mean of the 
is
![\[M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}\]](http://latex.artofproblemsolving.com/a/9/8/a9805b4fe344fe8975d89a778841e9abd9b214ab.png)
when 
and is given by the geometric mean of the when 
.
For any finite set of positive reals, 
, we have that 
implies 
and equality holds if and only if 
.
The Power Mean Inequality follows from the fact that 
together with Jensen's Inequality.
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