Power 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 $a_1, a_2, \ldots, a_n$ , the $k$ th power mean of the $a_i$ is

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

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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Power Mean Inequality

Inequality