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.

Statement

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Let   be the open unit disk in the complex plane   centered at the origin, and let   be a holomorphic map such that   and   on  .

Then   for all  , and  .

Moreover, if   for some non-zero   or  , then   for some   with  .[1]

Proof

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The proof is a straightforward application of the maximum modulus principle on the function

 

which is holomorphic on the whole of  , including at the origin (because   is differentiable at the origin and fixes zero). Now if   denotes the closed disk of radius   centered at the origin, then the maximum modulus principle implies that, for  , given any  , there exists   on the boundary of   such that

 

As   we get  .

Moreover, suppose that   for some non-zero  , or  . Then,   at some point of  . So by the maximum modulus principle,   is equal to a constant   such that  . Therefore,  , as desired.

Schwarz–Pick theorem

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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   be holomorphic. Then, for all  ,

 

and, for all  ,

 

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   can be made as follows:

Let   be holomorphic. Then, for all  ,

 

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  . Then, the map   is a holomorphic map from   onto  . Using the Schwarz–Pick theorem on this map, and finally simplifying the results by using the formula for  , we get the desired result. Also, for all  ,

 

If equality holds for either the one or the other expressions, then   must be a Möbius transformation with real coefficients. That is, if equality holds, then

 

with   and  .

Proof of Schwarz–Pick theorem

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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. Fix   and define the Möbius transformations

 

Since   and the Möbius transformation is invertible, the composition   maps   to   and the unit disk is mapped into itself. 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   tend to  .

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

De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of   at   in case   is injective; that is, univalent.

The Koebe 1/4 theorem provides a related estimate in the case that   is univalent.

See also

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References

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  1. ^ Theorem 5.34 in Rodriguez, Jane P. Gilman, Irwin Kra, Rubi E. (2007). Complex analysis : in the spirit of Lipman Bers ([Online] ed.). New York: Springer. p. 95. ISBN 978-0-387-74714-9.{{cite book}}: CS1 maint: multiple names: authors list (link)

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