Let $f(z)$ be a complex analytic function on the annulus $r_1\leq\abs{z}\leq r_3$ . Let $M(r)$ be the maximum of $\abs{f(z)}$ on the circle $\abs{z}=r$ . 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 as a function of $\log r$ .

The conclusion of the theorem can be restated as \begin{equation*} \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) \end{equation*}for any three concentric circles of radii $r_1<r_2<r_3$ .