# proof of Hadamard three-circle theorem

Let $f$ be holomorphic on a closed annulus $$. Let

$$s=\frac{\mathrm{log}{r}_{1}-\mathrm{log}r}{\mathrm{log}{r}_{2}-\mathrm{log}{r}_{1}}.$$ |

Let $M(r)={M}_{f}(r)={||f||}_{r}={\mathrm{max}}_{|z|=r}|f(z)|$. Then we have to prove that

$$\mathrm{log}M(r)\le (1-s)\mathrm{log}M({r}_{1})+s\mathrm{log}M({r}_{2}).$$ |

For this, let $\alpha $ be a real number; the function $\alpha \mathrm{log}|z|+\mathrm{log}|f(z)|$ is harmonic outside the zeros of $f$. Near the zeros of $f$ the above function has values which are large negative. Hence by the maximum modulus principle this function has its maximum on the boundary of the annulus, specifically on the two circles $|z|={r}_{1}$ and $|z|={r}_{2}$. Therefore

$$\alpha \mathrm{log}|z|+\mathrm{log}|f(z)|\le \mathrm{max}(\alpha \mathrm{log}{r}_{1}+\mathrm{log}M({r}_{1}),\alpha \mathrm{log}{r}_{2}+\mathrm{log}M({r}_{2}))$$ |

for all $z$ in the annulus. In particular, we get the inequality^{}

$$\alpha \mathrm{log}r+\mathrm{log}M(r)\le \mathrm{max}(\alpha \mathrm{log}{r}_{1}+\mathrm{log}M({r}_{1}),\alpha \mathrm{log}{r}_{2}+\mathrm{log}M({r}_{2})).$$ |

Now let $\alpha $ be such that the two values inside the parentheses on the right are equal, that is

$$\alpha =\frac{\mathrm{log}M({r}_{2})-\mathrm{log}M({r}_{1})}{\mathrm{log}{r}_{1}-\mathrm{log}{r}_{2}}.$$ |

Then from the previous inequality, we get

$$\mathrm{log}M(r)\le \alpha \mathrm{log}{r}_{1}+\mathrm{log}M({r}_{1})-\alpha \mathrm{log}r,$$ |

which upon substituting the value for $\alpha $ gives the result stated in the theorem.

## References

Lang, S.
*Complex analysis, Fourth edition*. Graduate Texts in Mathematics, 103.
Springer-Verlag, New York, 1999. xiv+485 pp. ISBN 0-387-98592-1

Title | proof of Hadamard three-circle theorem |
---|---|

Canonical name | ProofOfHadamardThreecircleTheorem |

Date of creation | 2013-03-22 15:56:02 |

Last modified on | 2013-03-22 15:56:02 |

Owner | Simone (5904) |

Last modified by | Simone (5904) |

Numerical id | 5 |

Author | Simone (5904) |

Entry type | Proof |

Classification | msc 30C80 |

Classification | msc 30A10 |