# Jensen's inequality

### From Wikimization

By definition is convex if and only if

whenever and are in the domain of .

It follows by induction on that if for then

(1)

Jensen's inequality says this:

If is a probability
measure on ,

is a real-valued function on ,

is integrable, and

is convex on the range
of then

(2)

**Proof 1:** By some limiting argument we can assume
that is simple (this limiting argument is the missing
detail).

That is, is the disjoint union of
and is constant on each .

Say and is the value of on . Then (1) and (2) say exactly the same thing. QED.

**Proof 2:**

Lemma. If and then

The lemma shows that has a right-hand
derivative at every point and that the graph of
lies above the "tangent" line through any point on the
graph with slope = the right derivative.

Say , let the right derivative of at , and let

The comment above says that for all in the domain of . So

D. Ullrich