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Three value ranges for symmetric self-mappings of the unit disc


Authors: Julia Koch and Sebastian Schleißinger
Journal: Proc. Amer. Math. Soc. 145 (2017), 1747-1761
MSC (2010): Primary 30C55, 30C80
DOI: https://doi.org/10.1090/proc/13350
Published electronically: October 26, 2016
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Abstract: Let $ \mathbb{D}$ be the unit disc and $ z_0\in \mathbb{D}.$ We determine the value range $ \{f(z_0)\,\vert\, f\in \mathcal {R}^\geq \}$, where $ \mathcal {R}^\geq $ is the set of holomorphic functions $ f:\mathbb{D}\to \mathbb{D}$ with $ f(0)=0$ and $ f'(0)\geq 0$ that have only real coefficients in their power series expansion around 0, and the smaller set $ \{f(z_0)\,\vert\, f\in \mathcal {R}^\geq ,$$ \text {$f$\ is typically real}\}.$

Furthermore, we describe a third value range $ \{ f(z_0) \,\vert\, f \in \mathcal {I}\}$, where $ \mathcal {I}$ consists of all univalent self-mappings of the upper half-plane $ \mathbb{H}$ with hydrodynamical normalization which are symmetric with respect to the imaginary axis.


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Julia Koch
Affiliation: Universität Würzburg, Institut für Mathematik, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
Email: julia.d.koch@mathematik.uni-wuerzburg.de

Sebastian Schleißinger
Affiliation: Università di Roma “Tor Vergata”, Dipartimento di Matematica, 00133 Roma, Italy
Email: schleiss@mat.uniroma2.it

DOI: https://doi.org/10.1090/proc/13350
Keywords: Value ranges, radial Loewner equation, chordal Loewner equation, univalent functions, typically real functions
Received by editor(s): February 16, 2016
Received by editor(s) in revised form: June 20, 2016, and June 25, 2016
Published electronically: October 26, 2016
Additional Notes: The second author was supported by the ERC grant “HEVO - Holomorphic Evolution Equations” No. 277691.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society