Abstract
In this chapter we describe some properties of univalent functions from the unit disc whose images are contained in \(\mathbb C\). The choice of the topics is based on the material we need in this book and not on the intrinsic relevance of the topics themselves inside the theory of univalent functions. We first prove the No Koebe Arcs Theorem, from which we obtain several results about pre-images of slits via univalent maps. Then we present the so-called Koebe Distortion Theorems. Finally, we consider families of univalent functions and prove the Carathéodory Kernel Convergence Theorem.
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Bracci, F., Contreras, M.D., Díaz-Madrigal, S. (2020). Univalent Functions. In: Continuous Semigroups of Holomorphic Self-maps of the Unit Disc. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-36782-4_3
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DOI: https://doi.org/10.1007/978-3-030-36782-4_3
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Publisher Name: Springer, Cham
Print ISBN: 978-3-030-36781-7
Online ISBN: 978-3-030-36782-4
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