부산대학교 도서관

What conditions on the entries of three $n$ by $m$ real matrices $X$, $Y$ and $W$ guarantee that the inequality $$ \sum_{i=1}^{n}\sum_{j=1}^{m}w_{ij}\phi(x_{ij})\ge \sum_{i=1}^{n}\sum_{j=1}^{m}w_{ij}\phi(y_{ij}) $$ holds for all continuous convex functions $\phi$? \par This question was answered in the vector ($m=1$) case by L. Fuchs [Mat. Tidsskr. B. {\bf 1947} (1947), 53--54; MR0024480]. The authors give several results for the general matrix case where $n$ and $m$ are unrestricted. Their first result can be viewed as a straightforward generalization of Fuchs' theorem. Their second result is a necessary and sufficient condition for the above inequality to hold for all continuous convex functions $\phi$ stated in terms of a certain Green function $G$. The remainder of the paper discusses the estimation of $\sum_{i=1}^{n}\sum_{j=1}^{m}w_{ij}[\phi(x_{ij})-\phi(y_{ij})]$ from a number of different points of view.