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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Conformal mapping, convexity and total absolute curvature
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by Maria Kourou
Conform. Geom. Dyn. 22 (2018), 15-32
Published electronically: March 5, 2018


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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Bibliographic Information
  • Maria Kourou
  • Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece
  • MR Author ID: 1257461
  • Email:
  • 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:
  • MathSciNet review: 3770612