Inequalities for M-matrices and inverse M-matrices☆

In this paper, we establish some determinantal inequalities concerning M-matrices and inverse M-matrices. The main results are as follows: 1. If A=(aij) is either an n×n M-matrix or inverse M-matrix , then for any permutation i1,i2,…,in of {1, 2, … , n}, (a) detA⩽(∏i=1naii)∏s=2n1-|ai1i2⋯ais-1isaisi1|ai1i1ai2i2⋯aisis. (b) detA=∏i=1naii if and only if A is essentially triangular. 2. If A=(aij) is an n×n M-matrix, B=(bij) is an n×n inverse M-matrix , A∘B denotes the Hadamard product of A and B, then A∘B is an M-matrix, and for any permutation i1,i2,…,in of {1,2,…,n}, det(A∘B)⩾det(AB)∏s=2naisisdetA[i1,i2,…,is-1]detA[i1,…,is-1,is]+bisisdetB[i1,i2,…,is-1]detB[i1,…,is-1,is]-1.