Further Properties of Analytic Functions

Abstract

The Uniqueness Theorem (6.9) states that a non-constant analytic function in a region cannot be constant on any open set. Similarly, according to Proposition 3.7, |f| cannot be constant. Thus a non-constant analytic function cannot map an open set into a point or a circular arc. By applying the Maximum-Modulus Theorem, we can derive the following sharper result on the mapping properties of an analytic function.