J. von Neumann, Some matrix inequalities and metrization of matrix space, Tomsk Univ. Rev. vol 1 (1937), pp. 286--300.
In the above article, Neumann showed the following inequality.
Let $X,Y,Z$ be three Hermitian matrices of order $n$, with eigenvalues $$x_1\geq x_2\geq \cdots\geq x_n,\quad y_1\geq y_2\geq \cdots\geq y_n,\quad z_1\geq z_2\geq \cdots\geq z_n.$$ Assume that $X-Y=Z$. Then $$\max_{j_1<j_2<\cdots<j_n}\sum_{i=1}^k (x_{j_i}-y_{j_i})\leq \sum_{i=1}^k z_i,\quad 1\leq k\leq n.$$
I could not prove it. Also, I find not find the reference [J. von Neumann, Some matrix inequalities and metrization of matrix space, Tomsk Univ. Rev. vol 1 (1937), pp. 286--300].