Power Mean Inequality

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

Inequality

For real numbers $k_1,k_2$ and positive real numbers $a_1, a_2, \ldots, a_n$ , $k_1\ge k_2$ implies the $k_1$ th power mean is greater than or equal to the $k_2$ th.

Algebraically, $k_1\ge k_2$ implies that $\left( \frac{\sum\limits_{i=1}^n a_{i}^{k_1}}{n} \right) ^ {\frac{1}{k_1}}\ge \left( \frac{\sum\limits_{i=1}^n a_{i}^{k_2}}{n} \right) ^ {\frac{1}{k_2}}$

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

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Power Mean Inequality

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Inequality