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Complex Analysis/Schwarz's Lemma

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The Schwarz Lemma is a statement about the growth behavior of holomorphic functions on the unit disk.

Statement

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Let be the unit disk, and let be holomorphic with . Then the following hold:

  • for all
  • If or for some , then is a rotation, i.e., there exists a with such that for all .

Proof

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Define by

Then is continuous and therefore, by the Riemann Removability Theorem, also holomorphic. Let . By the Maximum Principle, for , we have:

As , it follows that , hence for all , proving the first two statements. If equality holds in either case, then has a local maximum in the interior of . By the Maximum Modulus Principle, must be constant. This constant has modulus , and the claim follows. See Fischer, p. 286.

See Also

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Translation and Version Control

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This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Lemma_von_Schwarz

  • Date: 01/08/2024