In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself.
The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove.
It is, however, one of the simplest results capturing the rigidity of holomorphic functions.
be the open unit disk in the complex plane
[1] The proof is a straightforward application of the maximum modulus principle on the function which is holomorphic on the whole of
is differentiable at the origin and fixes zero).
denotes the closed disk of radius
centered at the origin, then the maximum modulus principle implies that, for
So by the maximum modulus principle,
A variant of the Schwarz lemma, known as the Schwarz–Pick theorem (after Georg Pick), characterizes the analytic automorphisms of the unit disc, i.e. bijective holomorphic mappings of the unit disc to itself: Let
, The expression is the distance of the points
in the Poincaré metric, i.e. the metric in the Poincaré disc model for hyperbolic geometry in dimension two.
The Schwarz–Pick theorem then essentially states that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric.
If equality holds throughout in one of the two inequalities above (which is equivalent to saying that the holomorphic map preserves the distance in the Poincaré metric), then
must be an analytic automorphism of the unit disc, given by a Möbius transformation mapping the unit disc to itself.
An analogous statement on the upper half-plane
, This is an easy consequence of the Schwarz–Pick theorem mentioned above: One just needs to remember that the Cayley transform
maps the upper half-plane
conformally onto the unit disc
Using the Schwarz–Pick theorem on this map, and finally simplifying the results by using the formula for
, If equality holds for either the one or the other expressions, then
must be a Möbius transformation with real coefficients.
The proof of the Schwarz–Pick theorem follows from Schwarz's lemma and the fact that a Möbius transformation of the form maps the unit circle to itself.
and define the Möbius transformations Since
and the Möbius transformation is invertible, the composition
Thus we can apply Schwarz's lemma, which is to say Now calling
(which will still be in the unit disk) yields the desired conclusion To prove the second part of the theorem, we rearrange the left-hand side into the difference quotient and let
De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of
The Koebe 1/4 theorem provides a related estimate in the case that
This article incorporates material from Schwarz lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.