In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Schwarz lemma, named after
Hermann Amandus Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.
Life
Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kumme ...
, is a result in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
about
holomorphic functions
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
from the
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose di ...
to itself. The
lemma is less celebrated than deeper theorems, such as the
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected space, simply connected open set, open subset of the complex plane, complex number plane C which is not all of C, then there exists a biholomorphy ...
, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.
Statement
Let
be the open
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose di ...
in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
centered at the
origin
Origin(s) or The Origin may refer to:
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* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
, and let
be a
holomorphic map
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivat ...
such that
and
on
.
Then
for all
, and
.
Moreover, if
for some non-zero
or
, then
for some
with
.
[Theorem 5.34 in ]
Proof
The proof is a straightforward application of the
maximum modulus principle
In mathematics, the maximum modulus principle in complex analysis states that if ''f'' is a holomorphic function, then the modulus , ''f'' , cannot exhibit a strict local maximum that is properly within the domain of ''f''.
In other words, eit ...
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
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
holomorphic map
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivat ...
pings 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
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry ...
, i.e. the metric in the Poincaré disc model for
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
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
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
mapping the unit disc to itself.
An analogous statement on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
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 In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The 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
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
with real coefficients. That is, if equality holds, then
:
with
and
.
Proof of Schwarz–Pick theorem
The proof of the Schwarz–Pick theorem follows from Schwarz's lemma and the fact that a
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
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
.
Further generalizations and related results
The
Schwarz–Ahlfors–Pick theorem
In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model.
The Schwarz–Pick lemma states that every holomorphic function from the unit disk ''U'' t ...
provides an analogous theorem for hyperbolic manifolds.
De Branges' theorem In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It ...
, 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
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
; that is,
univalent.
The
Koebe 1/4 theorem In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following:
Koebe Quarter Theorem. The image of an injective analytic function f:\mathbf\to\mathbb from the unit disk
\mathbf onto a subset of the complex plane con ...
provides a related estimate in the case that
is univalent.
See also
*
Nevanlinna–Pick interpolation In complex analysis, given ''initial data'' consisting of n points \lambda_1, \ldots, \lambda_n in the complex unit disc \mathbb and ''target data'' consisting of n points z_1, \ldots, z_n in \mathbb, the Nevanlinna–Pick interpolation problem is ...
References
* Jurgen Jost, ''Compact Riemann Surfaces'' (2002), Springer-Verlag, New York. ''(See Section 2.3)''
*
{{PlanetMath attribution, title=Schwarz lemma, id=3047
Riemann surfaces
Lemmas in analysis
Theorems in complex analysis
Articles containing proofs