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]] k and [[positive]] real numbers <math>a_1, a_2, \ldots, a_n</math>, the | + | 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 |
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=== Inequality === | === 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>\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 
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.
