Schwarz's Lemma

Theorem
Let $D$ be the unit disk centred at $0$.

Let $f: D \to \C$ be a holomorphic function.

Let $\map f 0 = 0$ and $\cmod {\map f z} \le 1$ for all $z \in D$.

Then $\cmod {\map {f'} 0} \le 1$, and $\cmod {\map f z} \le \cmod z$ for all $z \in D$.

Proof
First a lemma:

Lemma
Let $g : D \to \C$ be a complex function such that:


 * $\map g z = \begin {cases} \frac {\map f z} z & z \ne 0 \\ \map {f'} 0 & z = 0\end {cases}$

By the lemma, $g$ is holomorphic on $D$.

Let $r$ be a real number with $0 < r < 1$.

Let $C_r$ be the closed disk of radius $r$ centred at $0$.

Since $C_r$ is compact, by Continuous Function on Compact Space is Bounded, $g$ attains its maximum on $C_r$.

Let $\alpha \in C_r$ be a point where this maximum is attained.

If $g$ is constant, then we can pick $\alpha$ to be any point on $C_r$.

If $g$ is non-constant, then by the maximum modulus principle, $\alpha$ necessarily lies on the boundary of $C_r$.

That is, $\cmod \alpha = r$ if $g$ is non-constant.

If $g$ is constant, pick $\alpha$ such that $\cmod \alpha = r$ to avoid splitting into cases.

With that, for all $z \in C_r$ we have:


 * $\cmod {\map g z} \le \cmod {\map g \alpha}$

Since $0$ lies in the interior of $C_r$, we must have $\alpha \ne 0$, so:

So, for all $z \in C_r$ we have:


 * $\cmod {\map g z} \le \dfrac 1 r$

there exists $\omega \in D$ with:


 * $\cmod \omega < 1$
 * $\cmod {\map g \omega} = 1 + \varepsilon > 1$

Note that we then have:


 * $0 < \dfrac 1 {1 + \varepsilon} < 1$

Note that on $C_R$ with $R > \map \max {\dfrac 1 {1 + \varepsilon}, \cmod \omega}$, we should have:


 * $\cmod {\map g z} \le \dfrac 1 R < 1 + \varepsilon$

However, we have $\omega \in C_R$ and:


 * $\cmod {\map g \omega} = 1 + \varepsilon$

a contradiction.

So, it must be the case that:


 * $\cmod {\map g z} \le 1$

for all $z \in D$.

We have:


 * $\cmod {\map {f'} 0} = \cmod {\map g 0} \le 1$

Note that:


 * $\cmod {\map f 0} = 0$

so certainly:


 * $\cmod {\map f z} \le \cmod z$

for $z = 0$.

Take $z \in D \setminus \set 0$, from:


 * $\cmod {\map g z} \le 1$

we have:


 * $\cmod {\dfrac {\map f z} z} \le 1$

that is:


 * $\cmod {\map f z} \le \cmod z$

hence the result.