Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A∘B-1) for the Hadamard product of A and B-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A∘B) for nonnegative matrices A and B.