Power Mean Inequality
The Power Mean Inequality is a generalized form of the multi-variable Arithmetic Mean-Geometric Mean 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.
We prove by cases:
2. for with
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 .
For , As the function is concave for all , by Jensen's Inequality, For , becomes convex as , so the inequality sign when applying Jensen's Inequalitythe inequality is flipped. Thus, the inequality sign in is flipped, but as , is a decreasing function, so the inequality sign is flipped again after applying , resulting in as desired.