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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Conformal mapping, convexity and total absolute curvature
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by Maria Kourou PDF
Conform. Geom. Dyn. 22 (2018), 15-32 Request permission

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.
References
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Additional Information
  • Maria Kourou
  • Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
  • MR Author ID: 1257461
  • Email: mkouroue@math.auth.gr
  • 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
  • © Copyright 2018 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 22 (2018), 15-32
  • MSC (2010): Primary 30C45, 30C35
  • DOI: https://doi.org/10.1090/ecgd/317
  • MathSciNet review: 3770612