Hadamard three-circle theorem

Let $f(z)$ be a complex analytic function on the annulus $r_{1}\leq\left\lvert z\right\rvert\leq r_{3}$ . Let $M(r)$ be the maximum of $\left\lvert f(z)\right\rvert$ on the circle $\left\lvert z\right\rvert=r$ . Then $\log M(r)$ is a convex function of $\log r$ . Moreover, if $f(z)$ is not of the form $cz^{\lambda}$ for some $\lambda$ , then $\log M(r)$ is a strictly convex (http://planetmath.org/ConvexFunction) as a function of $\log r$ .

The conclusion of the theorem can be restated as

\log\frac{r_{3}}{r_{1}}\log M(r_{2})\leq\log\frac{r_{3}}{r_{2}}\log M(r_{1})+% \log\frac{r_{2}}{r_{1}}\log M(r_{3})

for any three concentric circles of radii $r_{1}<r_{2}<r_{3}$ .

Title	Hadamard three-circle theorem
Canonical name	HadamardThreecircleTheorem
Date of creation	2013-03-22 14:10:45
Last modified on	2013-03-22 14:10:45
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	7
Author	bbukh (348)
Entry type	Theorem
Classification	msc 30A10
Classification	msc 30C80
Related topic	MaximumPrinciple
Related topic	LogarithmicallyConvexFunction
Related topic	HardysTheorem