We introduce the “mth order Carathéodory-Reiffen metric,” the “mth order Bergman metric” and the “mth order modified Bergman metric” on M. Here M is a complex manifold which is ample in a suitable sense. These “metrics” are defined on T ( M ) T(M) and they are intrinsic. They arise as solutions of maximum problems. The first orders of these “metrics” (except for the modified Bergman metric) are the corresponding familiar metrics. All these metrics are biholomorphically invariant. We establish a chain of inequalities between them. This generalizes an earlier result of Hahn, proved by different methods.