On functions that preserve M-matrices and inverse M-matrices

In earlier works, authors such as Varga, Micchelli and Willoughby, Ando, and Fiedler and Schneider have studied and characterized functions which preserve the M-matrices or some subclasses of the M-matrices, such as the Stieltjes matrices. Here we characterize functions which either preserve the inverse M-matrices or map the inverse M-matrices to the M-matrices. In one of our results we employ the theory of Pick functions to show that if A and B are inverse M-matrices such that B ^?1 ≤ A ^?1, then (B+tI)?1 ≤ (A+tI)?1, for all t?≥?0.