# 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}.

Title Hadamard three-circle theorem HadamardThreecircleTheorem 2013-03-22 14:10:45 2013-03-22 14:10:45 bbukh (348) bbukh (348) 7 bbukh (348) Theorem msc 30A10 msc 30C80 MaximumPrinciple LogarithmicallyConvexFunction HardysTheorem