Schwarz lemma

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

 about holomorphic functions from the open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 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

Let be the open unit disk in the complex plane

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 C centered at the origin

Origin

Origin, origins, or original may refer to:- Beginning of the universe and life :* Abiogenesis, the study of how life on Earth arose from inanimate matter* Cosmogony, any theory concerning the origin of the universe...

 and let be a holomorphic map such that and for all .

Then, for all and

Moreover, if for some or

then for some with

Proof

The proof is a straightforward application of the maximum modulus principle

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 true local maximum that is properly within the domain of f....

 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

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 r approaches 1 we get

Moreover, suppose that for some nonzero , 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 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 ƒ : D → D be holomorphic. Then, for all z₁, z₂ ∈ D,


and, for all z ∈ D,


The expression


is the distance of the points z₁, z₂ in the Poincaré metric

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 or Riemann surfaces.There are three equivalent...

, i.e. the metric in the Poincaré disc model for hyperbolic geometry

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

 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 ƒ 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 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

Cayley transform

In mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping in...

 W(z) = (z − i)/(z + i) maps the upper half-plane H conformally onto the unit disc D. Then, the map W o ƒ 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 ƒ must be a Möbius transformation with real coefficients. That is, if equality holds, then


with a, b, c, d being real numbers, 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 where |z₀| < 1 maps the unit circle to itself. Fix z₁ and define the Möbius transformations

Since M(z₁) = 0 and the Möbius transformation is invertible, the composition φ(ƒ(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 z₂ = M ⁻¹(z) (which will still be in the unit disk) yields the desired conclusion


To prove the second part of the theorem, we just let z₂ tend to z₁.

Further generalizations and related results

The Schwarz–Ahlfors–Pick theorem

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...

 provides an analogous theorem for hyperbolic manifolds.

De Branges' theorem

De Branges' theorem

In complex analysis, the Bieberbach conjecture or de Branges's theorem, posed by and proven by , states a necessary condition on a holomorphic function to map the open unit disk of the complex plane injectively to the complex plane....

, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of ƒ at 0 in case ƒ is injective; that is, univalent.

The Koebe 1/4 theorem

Koebe 1/4 theorem

In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states that the image of an injective analytic function f:\mathbb D\to\mathbb C from the unit disk \mathbb D onto a subset of the complex plane contains the disk whose center is f\, and whose radius is |f\,'|/4. The theorem is...

provides a related estimate in the case that ƒ is univalent.