矩阵凸函数的微分分析
Differential analysis of matrix convex functions
摘要：
我们分析矩阵凸函数的一个固定的顺序定义在一个实区间上的微分方法，而不是由克劳斯给出的均差的特征。我们得到的每一个顺序的条件矩阵凸性是必要的和局部充分的，他们允许我们证明之间的差距类的连续订单的矩阵凸函数的存在，并给出明确的例子中包含的功能类型的每个这些差距。给出的条件也是全局充分的矩阵凸性的二阶。最后，我们引入了一个分数变换，它连接了每个n阶矩阵单调函数的集合与n+1阶矩阵凸函数的集合。
英文摘要：
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.
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