Some new bounds on eigenvalues of the Hadamard product and the Fan product of matrices

Let A and B be nonsingular M-matrices. A lower bound on the minimum eigenvalue q(B∘A-1) for the Hadamard product of A-1 and B, and a lower bound on the minimum eigenvalue q(AB) for the Fan product of A and B are given. In addition, an upper bound on the spectral radius ρ(A∘B) of nonnegative matrices A and B is also obtained. These bounds improve several existing results in some cases and the estimating formulas are easier to calculate for they are only depending on the entries of matrices A and B.