A note on Newton and Newton-like inequalities for M-matrices and for Drazin inverses of M-matrices

In a recent paper Holtz showed that M-matrices satisfy Newton's inequalities and so do the inverses of nonsingular M-matrices. Since nonsingular M-matrices and their inverses display various types of monotonic behavior, monotonicity properties adapted for Newton's inequalities are examined for nonsingular M-matrices and their inverses. In the second part of the paper the problem of whether Drazin inverses of singular M-matrices satisfy Newton's inequalities is considered. In general the answer is no, but it is shown that they do satisfy a form of Newton-like inequalities. In the final part of the paper the relationship between the satisfaction of Newton's inequality by a matrix and by its principal submatrices of order one less is examined, which leads to a condition for the failure of Newton's inequalities for the whole matrix.