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

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.