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Power Mean Inequality: Difference between revisions

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=== The Mean ===
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]] k and [[positive]] real numbers <math>a_1, a_2, \ldots, a_n</math>, the kth 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>
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=== Inequality ===
=== Inequality ===


If <math>a < b</math> then <math>M(a) \leq M(b)</math> and equality holds if and only if <math>\displaystyle a_1 = a_2 = \ldots = a_n</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>\displaystyle 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]].

Revision as of 17:20, 4 August 2006

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

For a real number $k$ and positive real numbers $a_1, a_2, \ldots, a_n$, the $k$th power mean of the $a_i$ is

$M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}$

when $k \neq 0$ and is given by the geometric mean of the $a_i$ when $k = 0$.

Inequality

For any finite set of positive reals, $\{a_1, a_2, \ldots, a_n\}$, we have that $a < b$ implies $M(a) \leq M(b)$ and equality holds if and only if $\displaystyle a_1 = a_2 = \ldots = a_n$.

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

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