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
Jensen's Inequality is a mathematical concept that relates to the convexity of a function. In complex analysis, it is used to prove the subharmonicity of a function, while in measure theory, it is used to prove the monotonicity of integrals.
Complex analysis deals with functions of a complex variable, while measure theory deals with the integration of functions over a set. The use of Jensen's Inequality in these two fields reflects their different applications and approaches to mathematical problems.
No, Jensen's Inequality can only be applied to convex functions. A function is convex if the line segment connecting any two points on the function's graph lies above or on the graph itself.
Jensen's Inequality is closely related to the concepts of convexity and concavity, as well as the Mean Value Theorem. It can also be used to prove other important theorems in mathematics, such as the Cauchy-Schwarz Inequality and the Hölder's Inequality.
Jensen's Inequality has numerous applications in various fields of mathematics, including complex analysis, measure theory, and optimization. It also has practical applications in economics, finance, and engineering, making it a fundamental concept in modern mathematics.