# 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.