# Maximum Modulus Theorem¶

This is a celebrated results in complex analysis, it says that the maximum of non-constant analytics function occur on the boundary of a domain.

Maximum Modulus Theorem

Let \(f\) is analytic in a connected domain \(\Omega\), then the maximum value of \(\vert f(z) \vert\) occur on the boundary of the domain, i.e., \(\partial \Omega\).

**Proof:** In order to prove this theorem first we will prove this result for a disc stated as follows

Local Mean Value Theorem

If \(f\) is analytic on a disc of radius \(r\) with centre \(a\), such that \(\vert f(a) \vert \geq \vert f(z) \vert\), for all \(z\) in the disc, then \(f\) is constant on the disc.

## 1. Schwartz Lemma¶

Schwartz Lemma

Let \(\bf{D}\) be the unit disc and \(f: \bf{D}\to \bf{D}\) is analytic function then,

- \(\vert f(z) \vert \leq \vert z \vert\) , for all \(z \in \bf{D}\)
- \(\vert f'(0) \vert \leq 1\)
- if there exist a \(z_0\in \bf{D}\), such that \(\vert f(z_0) \vert = \vert z_0 \vert\) or \(\vert f'(0) = 1\), then \(f\) is a rotation.

Last modified on: 2023-01-04 23:20:05