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The Carathe'odory and Kobayashi distance functions on a bounded domain G in Cn have related infinitesimal forms. These are the Caratheodory and Kobayashi metrics. They are denoted by F(z, t) Oength of the tangent vector t at the point z). They are defined in terms of holomorphic mappings, from G to the unit disk for the Carathe'odory metric, and from the unit disk to G for the Kobayashi metric. We consider the boundary behavior of these metrics on strongly pseudoconvex domains in Cn with C2 boundary. t is fixed and z is allowed to approach a boundary point z0. The quantity F(z, t) d(z, 8G) is shown to have a finite limit. In addition, if t belongs to the complex tangent space to the boundary at z0, then this first limit is zero, and (F(z, t))2d(z, 8G) has a (nontangential) limit in which the Levi form appears. We prove an approximation theorem for bounded holomorphic functions which uses peak functions in a novel way. The proof was suggested by N. Kerzman. This theorem is used here in studying the boundary behavior of the Carathe'odory metric.

Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in 𝐶ⁿ with smooth boundary