# Hadamard three-circle theorem

Let $f(z)$ be a complex analytic function on the annulus^{}
${r}_{1}\le \left|z\right|\le {r}_{3}$. Let $M(r)$ be the maximum of
$\left|f(z)\right|$ on the circle $\left|z\right|=r$. Then $\mathrm{log}M(r)$ is a
convex function of $\mathrm{log}r$. Moreover, if $f(z)$ is not of the form $c{z}^{\lambda}$ for some $\lambda $, then $\mathrm{log}M(r)$ is a strictly convex (http://planetmath.org/ConvexFunction) as a function of $\mathrm{log}r$.

The conclusion^{} of the theorem can be restated as

$$\mathrm{log}\frac{{r}_{3}}{{r}_{1}}\mathrm{log}M({r}_{2})\le \mathrm{log}\frac{{r}_{3}}{{r}_{2}}\mathrm{log}M({r}_{1})+\mathrm{log}\frac{{r}_{2}}{{r}_{1}}\mathrm{log}M({r}_{3})$$ |

for any three concentric circles of radii $$.

Title | Hadamard three-circle theorem |
---|---|

Canonical name | HadamardThreecircleTheorem |

Date of creation | 2013-03-22 14:10:45 |

Last modified on | 2013-03-22 14:10:45 |

Owner | bbukh (348) |

Last modified by | bbukh (348) |

Numerical id | 7 |

Author | bbukh (348) |

Entry type | Theorem |

Classification | msc 30A10 |

Classification | msc 30C80 |

Related topic | MaximumPrinciple |

Related topic | LogarithmicallyConvexFunction |

Related topic | HardysTheorem |