Schwarz lemma

The Schwarz lemma states that a holomorphic map from the unit disk to itself fixing zero satisfies |f(z)| ≤ |z| and |f′(0)| ≤ 1. Equality at a nonzero point forces the map to be a rotation.

Maximum modulus principle: A nonconstant holomorphic function cannot attain a local maximum of its modulus inside its domain. Applied to f(z)/z, it bounds the map’s growth throughout the disk.
Unit disk: The set of complex numbers whose modulus is less than one. It is both the domain and codomain in the theorem’s standard form.
Riemann mapping theorem: A theorem stating that every simply connected proper domain in the complex plane is conformally equivalent to the unit disk. Its normalized maps use the lemma to establish uniqueness.
Liouville's theorem: A bounded entire function on the complex plane must be constant. It also turns boundedness into rigidity, but on the whole plane rather than the disk.
Schwarz–Pick theorem: A theorem bounding how holomorphic maps between unit disks contract the hyperbolic metric. It extends the lemma from maps fixing zero to arbitrary disk maps.
Complex derivative: The limit defining the rate of change of a complex function at a point. The theorem bounds the derivative magnitude at the fixed point.
Conformal map: A holomorphic map that preserves angles locally wherever its derivative is nonzero. The lemma constrains normalized conformal maps between disk domains.
Cauchy's estimates: Bounds on derivatives of holomorphic functions obtained from Cauchy’s integral formula. They offer a broader family of analytic bounds, while the lemma exploits a fixed zero.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Holomorphicity is the condition that makes the lemma’s rigidity possible.
Open mapping theorem: A nonconstant holomorphic function maps open sets to open sets. It helps explain why equality forces a rigid, rather than merely boundary-attaining, map.

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