Hermitian Finsler metrics and the Kobayashi metric

The problem of local equivalence of Hermitian Finsler metrics under holomorphic changes of coordinates is solved. On such a Finsler metric we find some differential conditions which imply that the Finsler metric is the Kobayashi metric of the underlying manifold (these conditions are satisfied if the metric is the Kobayashi metric on a bounded, strictly convex domain in Cⁿ with smooth boundary).