# Schwarz lemma

Let $$ be the open unit disk in the complex plane^{} $\u2102$. Let $f:\mathrm{\Delta}\to \mathrm{\Delta}$ be a holomorphic function^{} with $f(0)=0$.
Then $|f(z)|\le |z|$ for all $z\in \mathrm{\Delta}$, and $|{f}^{\prime}(0)|\le 1$. If the equality $|f(z)|=|z|$ holds for *any* $z\ne 0$ or $|{f}^{\prime}(0)|=1$, then $f$ is a rotation: $f(z)=az$ with $|a|=1$.

This lemma is less celebrated than the bigger guns (such as the Riemann mapping theorem^{}, which it helps prove); however, it is one of the simplest results capturing the “rigidity” of holomorphic functions. No result exists for real functions, of course.

Title | Schwarz lemma^{} |
---|---|

Canonical name | SchwarzLemma |

Date of creation | 2013-03-22 12:44:37 |

Last modified on | 2013-03-22 12:44:37 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 8 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 30C80 |