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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 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.

Schwarz Lemma.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, and .

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

^{ [1] }

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

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

As we get .

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.

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 *f* : **D** → **D** be holomorphic. Then, for all *z*_{1}, *z*_{2} ∈ **D**,

and, for all *z* ∈ **D**,

The expression

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:

Let

f:H→Hbe holomorphic. Then, for allz_{1},z_{2}∈H,

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**,

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

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

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 *z*_{1} and define the Möbius transformations

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

Now calling *z*_{2} = *M*^{−1}(*z*) (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 *z*_{2} tend to *z*_{1}.

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.

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

In complex analysis, the **Riemann mapping theorem** states that if *U* is a non-empty simply connected open subset of the complex number plane **C** which is not all of **C**, then there exists a biholomorphic mapping *f* from *U* onto the open unit disk

In mathematics, the **Cauchy–Schwarz inequality**, also known as the **Cauchy–Bunyakovsky–Schwarz inequality**, is a useful inequality in many mathematical fields, such as linear algebra, analysis, probability theory, vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics.

In mathematics, **Cauchy's integral formula**, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

In mathematics, **Green's theorem** gives the relationship between a line integral around a simple closed curve *C* and a double integral over the plane region *D* bounded by *C*. It is the two-dimensional special case of Stokes' theorem.

In complex analysis, **Liouville's theorem**, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant. Equivalently, non-constant holomorphic functions on have unbounded images.

In geometry and complex analysis, a **Möbius transformation** of the complex plane is a rational function of the form

The theory of functions of **several complex variables** is the branch of mathematics dealing with complex-valued functions

In mathematics, the **Schwarzian derivative**, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces.

In contexts including complex manifolds and algebraic geometry, a **logarithmic** differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.

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 or Riemann surfaces.

In mathematics, the **Schwarz–Ahlfors–Pick** theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model.

In potential theory, the **Poisson kernel** is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.

In complex analysis, a branch of mathematics, the **Koebe 1/4 theorem** states the following:

Koebe Quarter Theorem.The image of an injective analytic functionf:D→Cfrom the unit diskDonto a subset of the complex plane contains the disk whose center isf(0) and whose radius is |f′(0)|/4.

In complex analysis, given *initial data* consisting of points in the complex unit disc and *target data* consisting of points in , the **Nevanlinna–Pick interpolation problem** is to find a holomorphic function that interpolates the data, that is for all ,

In complex analysis of one and several complex variables, **Wirtinger derivatives**, named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables.

In mathematics, the **Beltrami equation**, named after Eugenio Beltrami, is the partial differential equation

In mathematics, the **Denjoy–Wolff theorem** is a theorem in complex analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers into itself. The result was proved independently in 1926 by the French mathematician Arnaud Denjoy and the Dutch mathematician Julius Wolff.

In complex analysis and geometric function theory, the **Grunsky matrices**, or **Grunsky operators**, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement. The **Grunsky inequalities** express boundedness properties of these matrices, which in general are contraction operators or in important special cases unitary operators. As Grunsky showed, these inequalities hold if and only if the holomorphic function is univalent. The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky matrix and its associated inequalities were originally formulated in a more general setting of univalent functions between a region bounded by finitely many sufficiently smooth Jordan curves and its complement: the results of Grunsky, Goluzin and Milin generalize to that case.

In mathematics, the **oscillator representation** is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the **oscillator semigroup** by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,**C**) corresponding to Möbius transformations that take the unit disk into itself.

In mathematics, the **Douady–Earle extension**, named after Adrien Douady and Clifford Earle, is a way of extending homeomorphisms of the unit circle in the complex plane to homeomorphisms of the closed unit disk, such that the extension is a diffeomorphism of the open disk. The extension is analytic on the open disk. The extension has an important equivariance property: if the homeomorphism is composed on either side with a Möbius transformation preserving the unit circle the extension is also obtained by composition with the same Möbius transformation. If the homeomorphism is quasisymmetric, the diffeomorphism is quasiconformal. An extension for quasisymmetric homeomorphisms had previously been given by Lars Ahlfors and Arne Beurling; a different equivariant construction had been given in 1985 by Pekka Tukia. Equivariant extensions have important applications in Teichmüller theory, for example they lead to a quick proof of the contractibility of the Teichmüller space of a Fuchsian group.

- ↑ 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.

- Jurgen Jost,
*Compact Riemann Surfaces*(2002), Springer-Verlag, New York. ISBN 3-540-43299-X*(See Section 2.3)* - S. Dineen (1989).
*The Schwarz Lemma*. Oxford. ISBN 0-19-853571-6.

*This article incorporates material from Schwarz lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

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