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, its derivative
is monotonically decreasing. We consider two cases.
If , then
If
, then
By the fundamental theorem of calculus, we have
Evaluating the integrals, each of the last two inequalities implies the same result:
so this is true for all
. Then we have
as desired.
Example
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)