For normalised generalized matrix functions f and g, we say that f dominates g if f(A)≥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.

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