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.
Facts
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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Sources
1. Schwarz Lemma (Wikipedia)
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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.
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