Schwarz Lemma & Maximum Modulus Principle

Schwarz Lemma

Let D=\{z:|z|<1\} be the open unit disk in the complex plane \mathbb{C} centered at the origin and let f:D\to D be a holomorphic map such that f(0)=0.

Then, |f(z)|\leq |z| for all z\in D and |f'(0)\leq 1.

Moreover, if |f(z)|=|z| for some non-zero z or |f'(0)|=1, then f(z)=az for some a\in\mathbb{C} with |a|=1.

Maximum modulus principle

Let f be a function holomorphic on some connected open subset D of the complex plane \mathbb{C} and taking complex values. If z_0 is a point in D such that |f(z_0)|\geq|f(z)| for all z in a neighborhood of z_0, then the function f is constant on D.

Informally, the modulus |f| cannot exhibit a true local maximum that is properly within the domain of f.


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