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 aconvex 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 $r_{1}<r_{2}<r_{3}$ .