Difference between revisions of "Power Mean Inequality"
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== Inequality == | == Inequality == | ||
| − | For [[real number]]s <math>k_1,k_2</math> and [[positive]] real numbers <math>a_1, a_2, \ldots, a_n</math>, <math>k_1\ge k_2</math> implies the <math>k_1</math>th [[power mean]]is greater than or equal to the <math>k_2</math>th. | + | For [[real number]]s <math>k_1,k_2</math> and [[positive]] real numbers <math>a_1, a_2, \ldots, a_n</math>, <math>k_1\ge k_2</math> implies the <math>k_1</math>th [[power mean]] is greater than or equal to the <math>k_2</math>th. |
Algebraically, <math>k_1\ge k_2</math> implies that | Algebraically, <math>k_1\ge k_2</math> implies that | ||
Revision as of 22:07, 20 December 2007
The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.
Inequality
For real numbers 
and positive real numbers 
, 
implies the 
th power mean is greater than or equal to the 
th.
Algebraically, implies that
![\[\left( \frac{\sum\limits_{i=1}^n a_{i}^{k_1}}{n} \right) ^ {\frac{1}{k_1}}\ge \left( \frac{\sum\limits_{i=1}^n a_{i}^{k_2}}{n} \right) ^ {\frac{1}{k_2}}\]](http://latex.artofproblemsolving.com/5/1/0/51029c86b6bad459ea65e48067cff291106b0d1b.png)
The Power Mean Inequality follows from the fact that 
(where 
is the 
th power mean) together with Jensen's Inequality.
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