Inequalities between intrinsic metrics

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 $T(M)$ 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.