# Can anyone explain me what is AM-GM inequality?

Chetan Mandayam Nayakar
312 Points
10 years ago

The most basic arithmetic mean-geometric mean (AM-GM) inequality states simply that if and are nonnegative real numbers, then , with equality if and only if . The last phrase with equality...' means two things: first, if , then (obvious); and second, if for some , then . It follows that if and , then inequality is strict: .

Here is a one-line proof of the AM-GM inequality for two variables:

The AM-GM inequality generalizes to nonnegative numbers.

AM-GM inequality:
If , then

with equality if and only if

.