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