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 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
Say
Let be the right derivative of
at
, and let
The bullets above say for
all
in the domain of
So
David C. Ullrich