Power Mean Inequality

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

\[\sqrt[k_1]{\frac{a_{1}^{k_1}+a_{2}^{k_1}+\cdots +a_{n}^{k_1}}{n}}\ge \sqrt[k_2]{\frac{a_{1}^{k_2}+a_{2}^{k_2}+\cdots +a_{n}^{k_2}}{n} \right)}}\] (Error compiling LaTeX. ! Missing } inserted.)

which can be written more concisely as

\[\sqrt[k_1]{\frac{\sum\limits_{i=1}^n a_{i}^{k_1}}{n}}\ge \sqrt[k_2]{\frac{\sum\limits_{i=1}^n a_{i}^{k_2}}{n} \right)}}\] (Error compiling LaTeX. ! Missing } inserted.)

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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