Abstract
This short chapter is devoted to the Schwarz lemma, which is a simple consequence of the power series expansion and the maximum principle. The Schwarz lemma is proved in Section 1, and it is used in Section 2 to determine the conformal self-maps of the unit disk. In Section 2 we formulate the Schwarz lemma to be invariant under the conformal self-maps of the unit disk, thereby obtaining Pick’s lemma. This leads in Section 3 to the hyperbolic metric and hyperbolic geometry of the unit disk.
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© 2001 Springer Science+Business Media New York
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Gamelin, T.W. (2001). The Schwarz Lemma and Hyperbolic Geometry. In: Complex Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21607-2_9
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DOI: https://doi.org/10.1007/978-0-387-21607-2_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-95069-3
Online ISBN: 978-0-387-21607-2
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