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

Juan Arango; Hugo Arbeláez; Diego Mejía

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arXiv · 1912.03190

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

Juan Arango·Hugo Arbeláez·Diego Mejía

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Abstract

Let the function $\varphi$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $\varphi (\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $\varphi$, $$|D_{\varphi}(z)|:=\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}.$$ In particular, we show how the Schwarzian derivative of $\varphi$ reveals the nature of the critical points.

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Juan Arango, Hugo Arbeláez, Diego Mejía. 2019-12-06. Level sets of the Hyperbolic Derivative for analytic self-maps of the unit disk. https://arxiv.org/abs/1912.03190

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