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Conformal Geometry and Dynamics

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Conformal mapping, convexity and total absolute curvature


Author: Maria Kourou
Journal: Conform. Geom. Dyn. 22 (2018), 15-32
MSC (2010): Primary 30C45, 30C35
DOI: https://doi.org/10.1090/ecgd/317
Published electronically: March 5, 2018
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Abstract: Let $ f$ be a holomorphic and locally univalent function on the unit disk $ \mathbb{D}$. Let $ C_r$ be the circle centered at the origin of radius $ r$, where $ 0<r <1$. We will prove that the total absolute curvature of $ f(C_r)$ is an increasing function of $ r$. Moreover, we present inequalities involving the $ \textup {L}^p$-norm of the curvature of $ f(C_r)$. Using the hyperbolic geometry of $ \mathbb{D}$, we will prove an analogous monotonicity result for the hyperbolic total curvature. In the case where $ f$ is a hyperbolically convex mapping of $ \mathbb{D}$ into itself, we compare the hyperbolic total curvature of the curves $ C_r$ and $ f(C_r)$ and show that their ratio is a decreasing function. The last result can also be seen as a geometric version of the classical Schwarz Lemma.


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Additional Information

Maria Kourou
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
Email: mkouroue@math.auth.gr

DOI: https://doi.org/10.1090/ecgd/317
Keywords: Total absolute curvature, hyperbolic convexity, conformal mapping, convexity
Received by editor(s): June 29, 2017
Received by editor(s) in revised form: November 23, 2017, and January 25, 2018
Published electronically: March 5, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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