Jensen's Inequality
Jensen's Inequality is an inequality discovered by Danish mathematician Johan Jensen in 1906.
Inequality
Let be a convex function of one real variable. Let
and let
satisfy
. Then
If is a Concave Function, we have:
Proof
We only prove the case where is concave. The proof for the other case is similar.
Let .
As
is concave, then its derivative
is monotonically decreasing. We consider two cases.
If , then
If
, then
By the fundamental theorem of calculus, we have
Evaluating the integrals, the last two inequalities both result in
as desired.
One of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Take (verify that
and
) and
. You'll get
. Similarly, arithmetic mean-geometric mean inequality can be obtained from Jensen's inequality by considering
.
Problems
Introductory
Prove AM-GM using Jensen's Inequality
Intermediate
- Prove that for any
, we have
.
- Show that in any triangle
we have
Olympiad
- Let
be positive real numbers. Prove that
(Source)