On some new bounds for eigenvalues of the Hadamard product and the Fan product of matrices

Let A and B be nonnegative matrices. A new upper bound on the spectral radius ρ(A∘B) is obtained. Meanwhile, a new lower bound on the smallest eigenvalue q(A★B) for the Fan product, and a new lower bound on the minimum eigenvalue q(B∘A-1) for the Hadamard product of B and A-1 of two nonsingular M-matrices A and B are given. These bounds improve some results of Li et al. [9] and generalize some corresponding results. The new estimating formulae are only depending on the entries of matrices A and B, therefore, they are easy to calculate.