Level curves for analytic self-maps of the unit disk (Q2329128)

Let \(\varphi:{\mathbb D}\to {\mathbb D}\) be a holomorphic function, where \({\mathbb D}\) denotes the open unit disk in the complex plane. The authors study the global behavior, the structure of the level sets, and the singular points of the function \[ \mu(z)=\frac{1-|\varphi|(z)|^2}{1-|z|^2}. \] It is obtained in particular that \(\mu\) is subharmonic and that the sets \(\{z\in {\mathbb D}:\mu(z)<\lambda\}\), \(\lambda\in (0,\infty)\), are star-like with respect to the origin.