# Power Mean Inequality

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

## Inequality

For positive real numbers and positive real weights with sum , the power mean with exponent , where , is defined by

The Power Mean Inequality states that for all real numbers and , if . In particular, for nonzero and , and equal weights (i.e. ), if , then

Considering the limiting behavior, we also have , and .

The Power Mean Inequality follows from Jensen's Inequality.

## Proof

We prove by cases:

1. for

2. for with

Case 1:

Note that As is concave, by Jensen's Inequality, the last inequality is true, proving . By replacing by , the last inequality implies as the inequality signs are flipped after multiplication by .

Case 2:

For , As the function is concave for all , by Jensen's Inequality, For , becomes convex as , so the inequality sign when applying Jensen's Inequality is flipped. Thus, the inequality sign in is flipped, but as , is a decreasing function, the inequality sign is flipped again after applying , resulting in as desired.