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Conformal Geometry and Dynamics

ISSN 1088-4173



Conformal mapping, convexity and total absolute curvature

Author: Maria Kourou
Journal: Conform. Geom. Dyn. 22 (2018), 15-32
MSC (2010): Primary 30C45, 30C35
Published electronically: March 5, 2018
MathSciNet review: 3770612
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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 $\mathrm {L}^p$-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
MR Author ID: 1257461

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