Difference between revisions of "Power Mean Inequality"

Revision as of 12:47, 11 July 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 kth 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 $k = 0$ .

If $a < b$ then $M(a) \leq M(b)$ 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.

@@ Line 1: / Line 1: @@
 === 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.
-The kth "Power Mean", with exponent k and a series (a_i) of positive real numbers is ,
+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
 :<math>
 M(k) = \left( \frac{\sum_{i=1}^n a_{i}^k}{n} \right) ^ {\frac{1}{k}}
 </math>
+when <math>k \neq 0</math> and is given by the [[geometric mean]] of the
+<math>a_i</math> when <math>k = 0</math>.
-(The case k=0 is taken to be the geometic mean)
+=== 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>.
-If ''a'' < ''b'', then M(''a'') &le; M(''b''). Equality if and only if ''a''<sub>1</sub> = ''a''<sub>2</sub> = ... = ''a''<sub>''n''</sub>, following from <math>\frac{\partial M(t)}{\partial t}\geq 0</math> proved 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]].