Geodesic
On Completeness With Respect to the Carathéodory Metric*
Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 261-263

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Let X be a bounded region in the plane. Define where f is an analytic function on X bounded in modulus by 1. We call d the Carathéodory metric.In this note we give necessary and sufficient conditions to ensure that this metric be complete. 
Selby, M. A. On Completeness With Respect to the Carathéodory Metric*. Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 261-263. doi: 10.4153/CMB-1974-051-2
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