Hadamard three-circle theorem

Let f(z) be a complex analytic function on the annulusPlanetmathPlanetmath r1|z|r3. Let M(r) be the maximum of |f(z)| on the circle |z|=r. Then logM(r) is a convex function of logr. Moreover, if f(z) is not of the form czλ for some λ, then logM(r) is a strictly convex (http://planetmath.org/ConvexFunction) as a function of logr.

The conclusionMathworldPlanetmath of the theorem can be restated as


for any three concentric circles of radii r1<r2<r3.

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