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.