Hadamard three-circle theorem

In complex analysis, a branch of mathematics, the Hadamard three-circle theorem is a result about the behavior of holomorphic functions.

Let <math>f(z)<math> be a holomorphic function on the annulus

<math>r_1\leq\left| z\right| \leq r_3.<math>

Let <math>M(r)<math> be the maximum of <math>|f(z)|<math> on the circle <math>|z|=r.<math> Then, <math>\log M(r)<math> is a convex function of the logarithm <math>\log (r).<math> Moreover, if <math>f(z)<math> is not of the form <math>cz^\lambda<math> for some constants <math>\lambda<math> and <math>c<math>, then <math>\log M(r)<math> is strictly convex as a function of <math>\log (r).<math>

The conclusion of the theorem can be restated as

<math>(r_3-r_1)\log M(r_2)\leq (r_3-r_2)\log M(r_1)+(r_2-r_1)\log M(r_3)<math>

for any three concentric circles of radii <math>r_1

See also

• maximum principle
• logarithmically convex function
• Hardy's theorem

This article incorporates material from Hadamard three-circle theorem on PlanetMath, which is licensed under the GFDL.