Level sets of the Hyperbolic Derivative for analytic self-maps of the unit disk

2019/12/06 by Arango, Juan, Arbeláez, Hugo, Mejía, Diego
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.03190

Abstract

Let the function φ be holomorphic in the unit disk \mathbbD of the complex plane ℂ and let φ(\mathbbD)⊂ \mathbbD. We study the level sets and the critical points of the hyperbolic derivative of φ, |D_φ(z)|:=((1-|z|²)|φ'(z)|)/(1-|φ(z)|²). In particular, we show how the Schwarzian derivative of φ reveals the nature of the critical points.

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• Analytic mappings of the unit disk which almost preserve hyperbolic area
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• Hyperbolic curvature of holomorphic level curves
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