Differential analysis of matrix convex functions

We analyze matrix convex functions of a fixed order defined on a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of order n+1.