We give an estimate for the distance function related to the Kobayashi metric on a bounded strictly pseudoconvex domain with C 2 -smooth boundary. Our formula relates the distance function on the domain with the Carnot-Carathodory metric on the boundary. The estimate is precise up to a bounded additive term. As a corollary we conclude that the domain equipped with this distance function is hyperbolic in the sense of Gromov.
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Balogh et al. (2000) studied this question.