Maximum Modulus Theorem¶

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., \(\partial \Omega\).

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.

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.