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

TL;DR

This paper investigates the level sets and critical points of the hyperbolic derivative of holomorphic self-maps of the unit disk, revealing how the Schwarzian derivative characterizes critical point behavior.

Contribution

It introduces a detailed analysis of the hyperbolic derivative's level sets and critical points, linking them to the Schwarzian derivative for holomorphic self-maps.

Findings

Characterization of critical points via Schwarzian derivative

Description of level sets of the hyperbolic derivative

Insights into the structure of holomorphic self-maps

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.