For normalised generalized matrix functions f and g,
we say that f dominates g if
f(A)[ges ]g(A) for every
M-matrix A. We first demonstrate a finite set of test matrices
for any such inequality. Then, using results
from group representation theory, all comparisons among immanants in certain
classes are determined.
This work parallels ongoing research into gmf inequalities on positive
semidefinite matrices, for which no
finite set of test matrices is available. However, the inequalities for
the two classes are quite different, and
the test matrices permit more rapid progress in the M-matrix case. Just
as in the positive semidefinite case,
the gmf inequalities we prove may be used to verify previously unknown
determinantal inequalities for M-matrices,
such as the symmetrized Fischer inequalities recently proved in the positive
semidefinite case.