# 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 and positive real numbers , implies the th power mean is greater than or equal to the th.

Algebraically, implies that

which can be written more concisely as

The Power Mean Inequality follows from the fact that (where is the th power mean) together with Jensen's Inequality.

## Proof

Raising both sides to the th power, we get

We can see that the function is concave for all , and so we can apply Jensen's Inequality. Therefore,
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