Power Mean Inequality

The Mean

The power mean inequality is a generalized form of the multi-variable AM-GM inequality.

The kth "Power Mean", with exponent k and a series (a_i) of positive real numbers is ,

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

(The case k=0 is taken to be the geometic mean)

Inequality

If −∞ ≤ a < b ≤ ∞, then M(a) ≤ M(b). Equality if and only if a₁ = a₂ = ... = a_n, following from $\frac{\partial M(t)}{\partial t}\geq 0$ for −∞ ≤ t ≤ ∞, proved with Jensen's inequality.

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Power Mean Inequality

The Mean

Inequality