# Jensen's inequality

1. Dec 12, 2008

### lark

Is the Jensen's inequality in complex analysis related to the one in measure theory, or did Jensen just go around finding inequalities?
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2. Dec 12, 2008

### mathman

It looks like the complex analysis inequality is a special case, where ln|f| is the convex function of the measure theory theorem. By dividing by 2pi, the measure on the unit circle is normalized to 1.

3. Dec 13, 2008

### lark

Only ln |f| isn't convex on the real axis - exp is convex - and f is complex, not real.
It looks tantalizingly close, so I wonder if it can be twisted somehow.
Laura

4. Dec 13, 2008

### mathman

As I read the theorem, ln(x) has to be convex (which it is), not ln|f|. Ln corresponds to phi in the general theorem. The only requirement on |f| is that it be L1 with respect to the measure.

Last edited: Dec 13, 2008
5. Dec 17, 2008

### lark

Convex means that if you draw a line between 2 points on the graph of $$\phi$$
then the graph between those 2 points is below or on the line. Ln isn't convex but its inverse exp is.
Laura

6. Dec 17, 2008

### mathman

Convex can be convex down or convex up. The main idea is that a straight line connecting any two points on the curve does not cross the curve. For example, circles are convex.