非负矩阵元素和的两个不等式

TWO INEQUALITIES ON NON-NEGATIVE MATRIX

对2002年Yang Xiaojing发表于Linear Algebra and its Application上的矩阵不等式的证明做了简化,并将结论改进为:给定非负的m×n矩阵X=(xij)并且X≠0,则对任意实数p≥1,有mi=1(∑nj=1xij)p+mτ∑nj=1(∑mi=1xij)p(∑mi=1∑nj=1xij)p+(mn)τ∑mi=1∑nj=1xipj≥mτ+nτ(mn)τ+min(mτ,nτ),并给出了不等式中等号成立的充要条件.

Suppose X= （xij） be an m×n matrix with non-negative real entries, which are not all equal to O. Then for p≥1, τ=p-1,one derives n^r∑i=1^m（∑j=1^nxu）^p＋m^r∑j=1^n（∑i=1^mxij）^p/（∑i=1^m ∑j=1^nxij）^p＋（mn）^r∑i=1^m∑j=1^nxij^p≥m^r＋n^r/（mn）^r＋min（m^r,n^r）.the necessary and sufficient conditions are obtained when equality holds.