Level curves for analytic self-maps of the unit disk

Let the function φ be holomorphic in the unit disk D of the complex plane C and let φ ( D ) ⊂ D . We study the global behavior, the structure of the level sets and the singular points of the function μ ( z ) : = ( 1 - | φ ( z ) | 2 ) / ( 1 - | z | 2 ) . In particular we show that μ is subharmonic and that the sets { z ∈ D : μ ( z ) < λ } , 0 < λ < ∞ , are starlike with respect to the origin.