# Jensen's inequality

### From Wikimization

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<br>Jensen's inequality says this: <br>If <math>\,\mu\,</math> is a probability | <br>Jensen's inequality says this: <br>If <math>\,\mu\,</math> is a probability | ||

- | measure on <math>\,X\,</math>, <br><math>\,f\,</math> is a real-valued function on <math>\,X\,</math>, | + | measure on <math>\,X\,</math> , <br><math>\,f\,</math> is a real-valued function on <math>\,X\,</math> , |

<br><math>\,f\,</math> is integrable, and <br><math>\,\phi\,</math> is convex on the range | <br><math>\,f\,</math> is integrable, and <br><math>\,\phi\,</math> is convex on the range | ||

of <math>\,f\,</math> then | of <math>\,f\,</math> then |

## Revision as of 12:04, 26 July 2008

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