In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions defined on the open unit disk.

Let Δ = {z: | z | < 1} be the open unit disk in the complex plane C. Let be a holomorphic function with f(0)=0. Then

for all z in Δ, and . If the equality

holds for any z≠0, or

,

then f is a rotation: f(z) = az, with | a | = 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-Pick theorem

A variant of the Schwarz lemma can be stated that is invariant under a change of coordinates on the unit disk. This variant is known as the Schwarz-Pick theorem:

Let be holomorphic. Then, for all ,

and, for all

If equality holds for either the one or the other expression, then f must be a Möbius transformation, in which case both expressions are identities.

An analogous statement on the upper half-plane can be made as follows:

Let be holomorphic. Then, for all ,

and, for all

If equality holds for either the one or the other expression, then f must be a Möbius transformation with real coefficients, in which case both expressions are identities. That is, if equality holds, then

with a,b,c,d being real numbers, and ad - bc > 0.

Further generalizations

The Schwarz-Ahlfors-Pick theorem provides an analogous theorem for hyperbolic manifolds.

Louis De Branges' theorem is an important generalization.

References

• Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See Section 2.3)