Level curves for analytic self-maps of the unit disk

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 global behavior, the structure of the level sets and the singular points of the function

$$\begin{aligned} \mu (z):=(1-|\varphi (z)|^2)/(1-|z|^2). \end{aligned}$$

In particular we show that $\mu$ is subharmonic and that the sets $\{z\in {\mathbb {D}} : \mu (z)<\lambda \},0<\lambda <\infty$, are starlike with respect to the origin.