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

Abstract

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