Jensen's Inequality
Let be a convex function of one real variable. Let
and let
satisfy
. Then
The proof of Jensen's inequality is very simple. Since the graph of every convex function lies above its tangent line at every point, we can compare the function with the linear function
, whose graph is tangent to the graph of
at the point
. Then the left hand side of the inequality is the same for
and
, while the right hand side is smaller for
. But the inequality for
is an identity!
The simplest example of the use of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Take and
. You'll get
. Similarly, arithmetic mean-geometric mean inequality can be obtained from Jensen's inequality by considering
.