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.
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Bak, J., Newman, D.J. (2010). Further Properties of Analytic Functions. In: Complex Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7288-0_7
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DOI: https://doi.org/10.1007/978-1-4419-7288-0_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-7287-3
Online ISBN: 978-1-4419-7288-0
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