• DOI:10.4064/AP85-3-8
• Corpus ID: 122118647

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}
}

• L. Frerick, G. Schmieder
• Published 2005
• Mathematics
• Annales Polonici Mathematici

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 boundary of G consists of two continua and z0 ∈ G, then the Carathéodory discs {z ∈ G : c(z, z0) < r}, r ∈ (0, 1), are connected. Proof. It is enough to show that for an arbitrary z1 ∈ G there is a curve connecting z0 and z1 on which c(·, z0) is increasing. Since c is invariant under conformal maps, we may replace G by its image under the Möbius transformation z 7→ (z − z0)/(z − z1), and therefore we may assume that z0 = 0, z1 = ∞. Moreover there is some conformal mapping onto some radial slit domain which leaves 0 and ∞ fixed (see e.g. [2, Theorem IX.24]). Since rotations also fix 0 and ∞ we can arrange that ∂G is contained in {z ∈ C : R(z) ≤ 0}. Using the continuity of c it is enough to show that c(·, 0) : (0,∞) → (0, 1), t 7→ c(t, 0), is increasing. Let therefore t1 > t0 > 0 be arbitrary. Note that φ(z) = (z − t1)/(z − t0) maps the closed left halfplane onto the closed disc bounded by the circle through 1 = φ(∞) and t1/t0 = φ(0) > 1, which is symmetrical about the real axis. This shows that 1 < |φ(z)| < t1/t0 for each 2000 Mathematics Subject Classification: 30C75, 30C80.