Generalized matrix function inequalities on M-matrices

For normalised generalized matrix functions f and g, we say that f dominate s g if f(A) greater than or equal to g(A) for every M-matrix. A. We first d emonstrate a finite set of test matrices for any such inequality. Then, usi ng results from group representation theory. all comparisons among immanant s in certain classes are determined. This work parallels ongoing research i nto gmf inequalities on positive semidefinite matrices, for which no finite set of test matrices is available. However, the inequalities for the two c lasses are quite different, and the test matrices permit more rapid progres s in the M-matrix case. Just as in the positive semidefinite case, the gmf inequalities we prove may be used to verify previously unknown determinanta l inequalities for M-matrices, such as the symmetrized Fischer inequalities recently proved in the positive semidefinite case.