Difference between revisions of "Power Mean Inequality"

Revision as of 21:06, 20 December 2007

The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean Inequality.

Inequality

For real numbers $k_1,k_2$ and positive real numbers $a_1, a_2, \ldots, a_n$ , $k_1\ge k_2$ implies the $k_1$ th power meanis greater than or equal to the $k_2$ th.

Algebraically, $k_1\ge k_2$ implies that $\left( \frac{\sum\limits_{i=1}^n a_{i}^{k_1}}{n} \right) ^ {\frac{1}{k_1}}\ge \left( \frac{\sum\limits_{i=1}^n a_{i}^{k_2}}{n} \right) ^ {\frac{1}{k_2}}$

The Power Mean Inequality follows from the fact that $\frac{\partial M(t)}{\partial t}\geq 0$ (where $M(x)$ is the $t$ th power mean) together with Jensen's Inequality.

This article is a stub. Help us out by expanding it.

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 == 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 [[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
 <cmath>
-M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}
+\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}}
 </cmath>
-when <math>k \neq 0</math> and is given by the [[geometric mean]] of the  <math>a_i</math> when <math>k = 0</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> (where <math>M(x)</math> is the <math>t</math>th power mean) 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}}
 [[Category:Inequality]]
 [[Category:Theorems]]

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Difference between revisions of "Power Mean Inequality"

Revision as of 21:06, 20 December 2007

Inequality