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\(\hat f\left( \lambda \right)\) 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.
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Cima, J.A., Rung, D.C. Normal functions and a class of associated boundary functions. Israel J. Math. 4, 119–126 (1966). https://doi.org/10.1007/BF02937456
- Normal Function
- Unit Disk
- Meromorphic Function
- Jordan Curve
- Riemann Sphere