Difference between revisions of "Power Mean Inequality"
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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 | ||
| − | + | <cmath> | |
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}} | ||
| − | </ | + | </cmath> |
when <math>k \neq 0</math> and is given by the [[geometric mean]] of the | when <math>k \neq 0</math> and is given by the [[geometric mean]] of the | ||
| − | <math>a_i< | + | <math></math>a_i<math> when </math>k = 0<math></math>. |
| − | + | == Inequality == | |
| − | For any [[finite]] [[set]] of positive reals, <math>\{a_1, a_2, \ldots, a_n\}</math>, we have that <math>a < b</math> implies <math>M(a) \leq M(b)</math> and [[equality condition|equality]] holds if and only if <math> | + | For any [[finite]] [[set]] of positive reals, <math>\{a_1, a_2, \ldots, a_n\}</math>, we have that <math>a < b</math> implies <math>M(a) \leq M(b)</math> and [[equality condition|equality]] holds if and only if <math>a_1 = a_2 = \ldots = a_n</math>. |
The Power Mean Inequality follows from the fact that <math>\frac{\partial M(t)}{\partial t}\geq 0</math> together with [[Jensen's Inequality]]. | The Power Mean Inequality follows from the fact that <math>\frac{\partial M(t)}{\partial t}\geq 0</math> together with [[Jensen's Inequality]]. | ||
{{stub}} | {{stub}} | ||
| + | [[Category:Number theory]] | ||
| + | [[Category:Theorems]] | ||
Revision as of 21:09, 7 October 2007
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
![\[M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}\]](http://latex.artofproblemsolving.com/a/9/8/a9805b4fe344fe8975d89a778841e9abd9b214ab.png)
when 
and is given by the geometric mean of the
$$ (Error compiling LaTeX. Unknown error_msg)a_i
k = 0$$ (Error compiling LaTeX. Unknown error_msg).
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.
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