Schwarz Lemma - Proof

Proof

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

g(z) = \begin{cases} \frac{f(z)}{z}\, & \mbox{if } z \neq 0 \\ f'(0) & \mbox{if } z = 0, \end{cases}

which is holomorphic on the whole of D, including at the origin (because f is differentiable at the origin and fixes zero). Now if Dr = {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 Dr, there exists zr on the boundary of Dr such that

As r → 1 we get |g(z)| ≤ 1.

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.

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