Jensen's Inequality: Complex Analysis vs. Measure Theory

[lark]

Is the Jensen's inequality in complex analysis related to the one in measure theory, or did Jensen just go around finding inequalities?
See attachment for details.

 

[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.

 

[lark]

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.

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

 

[mathman]

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

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.

 

[lark]

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.

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

 

[mathman]

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

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.