# Jensen's inequality

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## Revision as of 16:50, 11 November 2009

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 a missing detail to this proof...)

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:

- has a right-hand derivative at every point, and
- the graph of lies above the "tangent" line through any point on the graph with slope equal to the right derivative.

Say

Let be the right derivative of at , and let

The bullets above say for all in the domain of . So

David C. Ullrich