en>Enyokoyama: /semisimple(ambiguity)->semisimple Lie algebra/

2014-05-13T01:18:23Z

semisimple(ambiguity)->semisimple Lie algebra

en>Pichpich: Disambiguated: adjoint representation → Adjoint representation of a Lie group

2012-06-24T12:57:49Z

Disambiguated: adjoint representation → Adjoint representation of a Lie group

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In [[mathematics]], the '''Schwarz lemma''', named after [[Hermann Amandus Schwarz]], is a result in [[complex analysis]] about [[holomorphic functions]] from the [[open set|open]] [[unit disk]] to itself. The lemma is less celebrated than stronger 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==
<blockquote>'''Schwarz Lemma.''' Let '''D''' = {''z'' : |''z''| < 1} be the open [[unit disk]] in the [[complex number|complex plane]] '''C''' centered at the [[origin (mathematics)|origin]] and let ''f'' : '''D''' → '''D''' be a [[holomorphic map]] such that ''f''(0) = 0.

Then, |''f''(''z'')| ≤ |''z''| for all ''z'' in '''D''' and |''f′''(0)| ≤ 1.

Moreover, if |''f''(''z'')| = |''z''| for some non-zero ''z'' or |''f′''(0)| = 1, then ''f''(''z'') = ''az'' for some ''a'' in '''C''' with |''a''| = 1.</blockquote>

'''Note.''' Some authors replace the condition ''f'' : '''D''' → '''D''' with |''f''(''z'')| ≤ 1 for all ''z'' in '''D''' (where ''f'' is still holomorphic in '''D'''). The two versions can be shown to be equivalent through an application of the [[maximum modulus principle]].

==Proof==
The proof is a straightforward application of the [[maximum modulus principle]] on the function

:<math>g(z) = \begin{cases}
\frac{f(z)}{z}\, & \mbox{if } z \neq 0 \\
f'(0) & \mbox{if } z = 0,
\end{cases}</math>

which is holomorphic on the whole of '''D''', including at the origin (because ''f'' is differentiable at the origin and fixes zero). Now if '''D'''''r'' = {''z'' : |''z''| ≤ ''r''} denotes the closed disk of radius ''r'' centered at the origin, then the maximum modulus principle implies that, for ''r'' < 1, given any ''z'' in '''D'''''r'', there exists ''z''''r'' on the boundary of '''D'''''r'' such that

:<math> |g(z)| \le |g(z_r)| = \frac{|f(z_r)|}{|z_r|} \le \frac{1}{r}.</math>

As ''r'' → 1 we get |''g''(''z'')| ≤ 1.

Moreover, suppose that |''f''(''z'')| = |''z''| for some non-zero ''z'' in '''D''', or |''f′''(0)| = 1. Then, |''g''(''z'')| = 1 at some point of '''D'''. So by the maximum modulus principle, ''g''(''z'') is equal to a constant ''a'' such that |''a''| = 1. Therefore, ''f''(''z'') = ''az'', as desired.

==Schwarz–Pick theorem==
A variant of the Schwarz lemma can be stated that is invariant under analytic automorphisms on the unit disk, i.e. [[bijective]] [[holomorphic]] mappings of the unit disc to itself. This variant is known as the '''Schwarz–Pick theorem''' (after [[Georg Pick]]):

Let ''f'' : '''D''' → '''D''' be holomorphic. Then, for all ''z''1, ''z''2 ∈ '''D''',

:<math>\left|\frac{f(z_1)-f(z_2)}{1-\overline{f(z_1)}f(z_2)}\right| \le \left|\frac{z_1-z_2}{1-\overline{z_1}z_2}\right|</math>

and, for all ''z'' ∈ '''D''',

:<math>\frac{\left|f'(z)\right|}{1-\left|f(z)\right|^2} \le \frac{1}{1-\left|z\right|^2}.</math>

The expression

:<math> d(z_1,z_2)=\tanh^{-1} \left|\frac{z_1-z_2}{1-\overline{z_1}z_2}\right| </math>

is the distance of the points ''z''1, ''z''2 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 ''f'' 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]] '''H''' can be made as follows:

<blockquote>Let ''f'' : '''H''' → '''H''' be holomorphic. Then, for all ''z''1, ''z''2 ∈ '''H''',

:<math>\left|\frac{f(z_1)-f(z_2)}{\overline{f(z_1)}-f(z_2)}\right|\le \frac{\left|z_1-z_2\right|}{\left|\overline{z_1}-z_2\right|}.</math>
</blockquote>

This is an easy consequence of the Schwarz–Pick theorem mentioned above: One just needs to remember that the [[Cayley transform]] ''W''(''z'') = (''z'' − ''i'')/(''z'' + ''i'') maps the upper half-plane '''H''' conformally onto the unit disc '''D'''. Then, the map ''W'' o ''f'' o ''W''−1 is a holomorphic map from '''D''' onto '''D'''. Using the Schwarz–Pick theorem on this map, and finally simplifying the results by using the formula for ''W'', we get the desired result. Also, for all ''z'' ∈ '''H''',

:<math>\frac{\left|f'(z)\right|}{\text{Im}(f(z))} \le \frac{1}{\text{Im}(z)}. </math>

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

:<math>f(z)=\frac{az+b}{cz+d}</math>

with ''a'', ''b'', ''c'', ''d'' ∈ '''R''', and ''ad'' − ''bc'' > 0.

==Proof of Schwarz–Pick theorem==
The proof of the Schwarz–Pick theorem follows from Schwarz's lemma and the fact that a [[Möbius transformation]] of the form

:<math>\frac{z-z_0}{\overline{z_0}z-1}, \qquad |z_0| < 1,</math>

maps the unit circle to itself. Fix ''z''1 and define the Möbius transformations

: <math>M(z)=\frac{z_1-z}{1-\overline{z_1}z}, \qquad \varphi(z)=\frac{f(z_1)-z}{1-\overline{f(z_1)}z}.</math>

Since ''M''(''z''1) = 0 and the Möbius transformation is invertible, the composition φ(''f''(''M''−1(''z''))) maps 0 to 0 and the unit disk is mapped into itself. Thus we can apply Schwarz's lemma, which is to say

:<math>\left |\varphi\left(f(M^{-1}(z))\right) \right|=\left|\frac{f(z_1)-f(M^{-1}(z))}{1-\overline{f(z_1)}f(M^{-1}(z))}\right| \le |z|.</math>

Now calling ''z''2 = ''M''−1(''z'') (which will still be in the unit disk) yields the desired conclusion

:<math>\left|\frac{f(z_1)-f(z_2)}{1-\overline{f(z_1)}f(z_2)}\right| \le \left|\frac{z_1-z_2}{1-\overline{z_1}z_2}\right|.</math>

To prove the second part of the theorem, we just let ''z''2 tend to ''z''1.

==Further generalizations and related results==
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 ''f'' at 0 in case ''f'' is [[injective]]; that is, [[univalent mapping|univalent]].

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

==References==
* Jurgen Jost, ''Compact Riemann Surfaces'' (2002), Springer-Verlag, New York. ISBN 3-540-43299-X ''(See Section 2.3)''
*{{cite book | author = S. Dineen | title = The Schwarz Lemma | publisher = Oxford | year = 1989 | isbn=0-19-853571-6 }}

{{PlanetMath attribution|title=Schwarz lemma|id=3047}}

[[Category:Riemann surfaces]]
[[Category:Lemmas]]
[[Category:Theorems in complex analysis]]
[[Category:Articles containing proofs]]

BF model - Revision history

en>Enyokoyama: /*semisimple(ambiguity)->semisimple Lie algebra*/

en>Pichpich: Disambiguated: adjoint representation → Adjoint representation of a Lie group

en>Enyokoyama: /semisimple(ambiguity)->semisimple Lie algebra/