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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,

  1. \(\vert f(z) \vert \leq \vert z \vert\) , for all \(z \in \bf{D}\)
  2. \(\vert f'(0) \vert \leq 1\)
  3. 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