The Schwarz Lemma states that if a holomorphic function maps the open unit disk into itself and fixes the origin, then the function's derivative at the origin is bounded by one in absolute value, and the function itself is bounded by the identity map unless it is a rotation. Named for Hermann Schwarz, it is a foundational result of complex analysis underlying the classification of automorphisms of the disk and the construction of the hyperbolic metric on it.
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StatementIf a holomorphic map from the open unit disk to itself fixes the origin, then its derivative at the origin is at most one in absolute value, and the map itself never sends a point farther from the origin than the point already was. 1 Classification
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Source Schwarz Lemma (Wikipedia)
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1. Schwarz Lemma (Wikipedia)
Wikimedia FoundationLead section, closing sentence
The Schwarz lemma has opened several branches of complex geometry, and become an essential tool in the use of geometric PDE methods in complex geometry.
Proved By: Hermann Schwarz, Lead paragraph
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise
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