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Schwarz lemma (Theorem)

Let $ \Delta = \{z: \vert z \vert < 1\}$ be the open unit disk in the complex plane $ \mathbb{C}$. Let $ f\colon\Delta\to\Delta$ be a holomorphic function with $ f(0)=0$. Then $ \vert f(z) \vert \le \vert z \vert$ for all $ z\in\Delta$, and $ \vert f'(0) \vert \le 1$. If the equality $ \vert f(z) \vert=\vert z \vert$ holds for any $ z\ne 0$ or $ \vert f'(0) \vert=1$, then $ f$ is a rotation: $ f(z)=az$ with $ \vert a \vert=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 similar result exists for real functions, of course.



"Schwarz lemma" is owned by Koro. [ full author list (3) | owner history (2) ]
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Cross-references: real functions, Riemann mapping theorem, rotation, equality, holomorphic function, complex plane, open unit disk
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This is version 5 of Schwarz lemma, born on 2002-06-05, modified 2006-09-13.
Object id is 3047, canonical name is SchwarzLemma.
Accessed 6740 times total.

Classification:
AMS MSC30C80 (Functions of a complex variable :: Geometric function theory :: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination)

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