一个关于非负矩阵的不等式

An Inequality on Non-negative Matrix

证明:若(xij)是一个元素不全为零的m×n非负矩阵,则当0<p<1时,有(m^(p-1)sum from i=1 to m ( ) r_i^p+n^(p-1) sum form j=1 to n ( ) c_j^p)/sum form to i=1 to m ( ) sum form to j=1 to n ( ) x_(ij)~p+(mn)^(p-1) sum form to i=1 to m ( ) sum form to j=1 to n ( )x_(ij)~p ≤m^(p-1)+n^(p-1)/(mn)^(p-1)+min(m^(p-1)),n^(p-1).这一结果是对2002年Yang Xiaojing发表在Linear Algebra and its Application上的当p1时此不等式的反向不等式的一个补充,使整个不等式得以更加完整.

In this note, we prove： if X = （xij） is an m × n matrix with non-negative real entries, which are not all equal to 0, then for 0 〈 p 〈 1, the following inequality holds ：m^p-1∑^mi=1rp＋n^p-1∑^nj=1cj^p/ （∑^mi=1∑^nj=1xij）^p＋（mn）^p-1∑^mi=1∑^nj=1xij^p≤m^p-1＋n^p-1/（mn）^p-1＋min（m^p-1,n^p-1） Our theorem complements a result of Xiaojing Yang [Linear Algebra Appl, 2002,348,41-47], who proved the converse of the inequality for p ≥ 1.