The discussion revolves around the relationship between Jensen's inequality in complex analysis and its counterpart in measure theory. Participants explore whether the inequalities are related or if they represent distinct findings by Jensen, focusing on the properties of convex functions involved.
Participants express differing views on the convexity of ln|f| and its implications for Jensen's inequality, indicating that multiple competing perspectives remain without a clear consensus.
There are unresolved questions regarding the assumptions about the convexity of ln|f| and the specific conditions under which Jensen's inequalities apply in both complex analysis and measure theory.
Only ln |f| isn't convex on the real axis - exp is convex - and f is complex, not real.mathman said: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.
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 said: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
Convex means that if you draw a line between 2 points on the graph of [tex]\phi[/tex]mathman said: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 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.lark said:Convex means that if you draw a line between 2 points on the graph of [tex]\phi[/tex]
then the graph between those 2 points is below or on the line. Ln isn't convex but its inverse exp is.
Laura