Some characterizations of finite Blaschke products of positive degree (Q1085306)

If a is in the unit disc \(\Delta\) let \(L_ a(z)=(a-z)/(1-\bar az)\). For any given finite set of numbers a(1),a(2),a(3),...,a(n) in \(\Delta\) and a constant \(\eta\) of modulus 1, \(\eta\prod^{n}_{k=1}L_{a(k)}(z)\) is said to be a finite Blaschke product of positive degree n. It is shown that the following four assertions are equivalent for a function f in the unit ball in \(H^{\infty}(\Delta)\) less constants of modulus 1. (1) f is a finite Blaschke product of positive degree. (2) A convergent sequence \((L_{f[S(k)(0)]}\circ f\circ S(k))_ 0^{\infty}\), where \((S(k))_ 0^{\infty}\) is a sequence of conformal automorphisms of \(\Delta\) tending to a constant of modulus 1, tends to a map of the form \(z\to \alpha z\) with \(| \alpha | =1.\) (3) The equality \(\lim_{| z| \to 1}(1-| z|^ 2)f'(z)(1-| f(z)|^ 2)^{-1}=1\) holds. (4) The function f has no asymptotic values in \(\Delta\) and the set \(\{\) \(z: f'(z)=0\}\) is finite.