Normal functions and a class of associated boundary functions

Letμ′ be the family of non-empty closed subsets of the Riemann sphere and Λ the family of continuous curves λ with values in the unit disk and lim _t→1 |λ(t)|=1. A meromorphic functionf in |z|<1 induces a mapping(hat f) from Λ intoμ′ by setting(hat fleft( lambda right)) equal to the cluster set off on λ. The authors show that if(hat f) is continuous then existence of an asymptotic value ate ^iθ implies the existence of an angular limit. Further if the spherical derivative off iso(1/(1?|z|)) then(hat f) is constant on every open disk in the space Λ.