Schwarz Lemma and Automorphisms of the Disk

The Schwarz lemma is a remarkably powerful rigidity result that constrains holomorphic self-maps of the unit disk. It leads to a complete description of the automorphism group of the disk.

The Schwarz Lemma

Theorem4.7Schwarz lemma

Let $f: \mathbb{D} \to \mathbb{D}$ be holomorphic with $f(0) = 0$ , where $\mathbb{D} = \{z : |z| < 1\}$ . Then:

$|f(z)| \leq |z|$ for all $z \in \mathbb{D}$ .
$|f'(0)| \leq 1$ .
If $|f(z_0)| = |z_0|$ for some $z_0 \neq 0$ or $|f'(0)| = 1$ , then $f(z) = e^{i\theta}z$ for some real $\theta$ (i.e., $f$ is a rotation).

Proof

Define $g(z) = f(z)/z$ for $z \neq 0$ and $g(0) = f'(0)$ . Then $g$ is holomorphic on $\mathbb{D}$ (the singularity at $0$ is removable since $f(0) = 0$ ).

For $|z| = r < 1$ : $|g(z)| = |f(z)|/r \leq 1/r$ . By the maximum modulus principle, $|g(z)| \leq 1/r$ for all $|z| \leq r$ . Letting $r \to 1^-$ : $|g(z)| \leq 1$ for all $z \in \mathbb{D}$ .

This gives $|f(z)| \leq |z|$ and $|f'(0)| = |g(0)| \leq 1$ .

If equality holds at any interior point, then $|g|$ attains its maximum in $\mathbb{D}$ , so $g$ is constant by the maximum modulus principle: $g(z) = e^{i\theta}$ , hence $f(z) = e^{i\theta}z$ . $\blacksquare$

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The Schwarz-Pick Lemma

Definition4.3Hyperbolic metric

The Poincare (hyperbolic) metric on $\mathbb{D}$ is

$ds = \frac{2|dz|}{1 - |z|^2}.$

The hyperbolic distance between $z, w \in \mathbb{D}$ is $d_h(z,w) = \tanh^{-1}\left|\frac{z-w}{1-\bar{w}z}\right|$ .

Theorem4.8Schwarz-Pick lemma

Every holomorphic function $f: \mathbb{D} \to \mathbb{D}$ is a contraction in the hyperbolic metric:

$\left|\frac{f(z) - f(w)}{1 - \overline{f(w)}f(z)}\right| \leq \left|\frac{z - w}{1 - \bar{w}z}\right|$

for all $z, w \in \mathbb{D}$ . Equivalently, $\frac{|f'(z)|}{1 - |f(z)|^2} \leq \frac{1}{1 - |z|^2}$ .

Equality holds if and only if $f$ is a Mobius transformation mapping $\mathbb{D}$ onto $\mathbb{D}$ .

Automorphisms of the Unit Disk

Definition4.4Automorphism

An automorphism of a domain $D$ is a bijective holomorphic map $f: D \to D$ (whose inverse is automatically holomorphic by the inverse function theorem).

Theorem4.9Automorphisms of the disk

Every automorphism of $\mathbb{D}$ has the form

$f(z) = e^{i\theta} \frac{z - a}{1 - \bar{a}z}$

for some $\theta \in \mathbb{R}$ and $a \in \mathbb{D}$ . The automorphism group $\text{Aut}(\mathbb{D})$ is isomorphic to $PSU(1,1) \cong PSL(2,\mathbb{R})$ .

ExampleMobius transformations of the disk

The map $\varphi_a(z) = \frac{z-a}{1-\bar{a}z}$ for $a = 1/2$ :

$\varphi_{1/2}(0) = -1/2$
$\varphi_{1/2}(1/2) = 0$ (it swaps $0$ and $a$ )
$\varphi_{1/2}$ is its own inverse: $\varphi_{1/2} \circ \varphi_{1/2} = \text{id}$

This is an involutory automorphism that interchanges $0$ and $1/2$ while preserving $\mathbb{D}$ .

Applications

ExampleFixed points of holomorphic self-maps

If $f: \mathbb{D} \to \mathbb{D}$ is holomorphic (not an automorphism) with $f(a) = a$ for some $a \in \mathbb{D}$ , then $|f'(a)| < 1$ strictly. To see this, conjugate by $\varphi_a$ to reduce to $g = \varphi_a \circ f \circ \varphi_a$ with $g(0) = 0$ , then apply the Schwarz lemma.

This is the basis of the Denjoy-Wolff theorem: iterates $f^n = f \circ \cdots \circ f$ converge to the unique fixed point (if it exists in $\mathbb{D}$ ).

RemarkConnections to hyperbolic geometry

The Schwarz-Pick lemma reveals that holomorphic self-maps of the disk are non-expanding in the hyperbolic metric. The automorphisms are precisely the isometries. This connects complex analysis to hyperbolic geometry and has deep consequences in geometric function theory and Teichmuller theory.