Connectedness of the Carathéodory discs for doubly connected domains

@article{Frerick2005ConnectednessOT,
  title={Connectedness of the Carath{\'e}odory discs for doubly connected domains},
  author={Leonhard Frerick and Gerald Schmieder},
  journal={Annales Polonici Mathematici},
  year={2005},
  volume={85},
  pages={281-282}
}
We prove that the Carathéodory discs for doubly connected domains in the complex plane are connected. Let G C be a domain and assume that no boundary component is a point. The Carathéodory distance c(z0, z1) between two points z0, z1 ∈ G is defined as sup |f(z0)|, where the supremum is taken over all f holomorphic in G whose modulus is bounded by 1 and which vanish at z1. The following theorem proves a conjecture of Pflug and Jarnicki ([1, p. 42]) in the doubly connected case: Theorem 1. If the… 
Carath\'{e}odory balls and proper holomorphic maps on multiply-connected planar domains
In this paper, we will establish the inequivalence of closed balls and the closure of open balls under the Carathéodory metric in some planar domains of finite connectivity greater than 2, and hence
Fridman Function, Injectivity Radius Function, and Squeezing Function
Very recently, the Fridman function of a complex manifold X has been identified as a dual of the squeezing function of X. In this paper, we prove that the Fridman function for certain hyperbolic

References

SHOWING 1-3 OF 3 REFERENCES
Invariant Distances and Metrics in Complex Analysis
As in the field of "Invariant Distances and Metrics in Complex Analysis" there was and is a continuous progress. This is the second extended edition of the corresponding monograph. This comprehensive
FRG E-mail: schmieder@mathematik.uni-oldenburg