# Problem with convex curve and Jensen inequality

• January 23rd 2008, 08:40 AM
chopet
Problem with convex curve and Jensen inequality
given a convex curve function f(x),

Jensen inequality states:
$E[ f(x) ] \geq f( E[x] )$

Preliminary:

Let l(x) be linear function below f(x). Since a convex function is the maximum of all linear functions that lie below it:

$f(x) \geq max\{l(x)\}$
>>> this equation I understand.

On the other hand, because f(x) is convex, there is always a value x for which f(x) = l(x), where l(x) is the tangent (or support line) to f(x).

So,
$f(x) \leq max\{l(x)\}$
>>> I don't understand at all.

So f(x) = max{l(x)}.
This is a relationship we are going to use to prove Jensen's inequality.

Can anyone explain the problem I have with the 2nd inequality first before I post the rest of the solution (which I can follow.) ?

Thanks.
• January 23rd 2008, 10:38 AM
earboth
Quote:

Originally Posted by chopet
given a convex curve function f(x),

...

On the other hand, because f(x) is convex, there is always a value x for which f(x) = l(x), where l(x) is the tangent (or support line) to f(x).

So,
$f(x) \leq max\{l(x)\}$
>>> I don't understand at all.

So f(x) = max{l(x)}.
...

Hello,

I've attached a sketch to demonstrate what this inequality means
• January 24th 2008, 04:40 AM
chopet
Hi, thanks for your sketch. I really appreciate it.
But the text says convex graph, and yours show a concave graph.

Anyway, I've reproduced the actual text (from google book) below.
The offending equation is circled in red.

http://www.geocities.com/kopenen/jensen.jpg