On two inequalities for the Hadamard product and the Fan product of matrices

If A and B are n×n nonsingular M-matrices, a lower bound on the smallest eigenvalue τ(AB) for the Fan product of A and B is given. In addition, using the estimate on the perron root of nonnegative matrices, we also obtain an upper bound on the spectral radius ρ(A∘B) for nonnegative matrices A and B. These bounds improve some existing results.