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)A=(aij) is either an n×nn×nM-matrix or inverse M  -matrix , then for any permutation i1,i2,…,ini1,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)A=(aij) is an n×nn×nM  -matrix, B=(bij)B=(bij) is an n×nn×n inverse M  -matrix , A∘BA∘B denotes the Hadamard product of A and B  , then A∘BA∘B is an M  -matrix, and for any permutation i1,i2,…,ini1,i2,…,in of {1,2,…,n}{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.