Difference between revisions of "Power Mean Inequality"
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The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] Inequality. | The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] Inequality. | ||
| − | For a [[real number]] <math>k</math> and [[positive]] real numbers <math>a_1, a_2, \ldots, a_n</math>, the <math>k</math>th power mean of the <math>a_i</math> is | + | For a [[real number]] <math>k</math> and [[positive]] real numbers <math>a_1, a_2, \ldots, a_n</math>, the <math>k</math>''th power mean'' of the <math>a_i</math> is |
| − | :<math> | + | :<math>\displaystyle |
M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}} | M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}} | ||
</math> | </math> | ||
Revision as of 18:20, 4 August 2006
The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.
For a real number 
and positive real numbers 
, the th power mean of the 
is
when 
and is given by the geometric mean of the
when 
.
Inequality
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.
