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 as a function of $\log r$.

The conclusion of the theorem can be restated as

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