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 {ring operator} A^{- 1}) for the Hadamard product of A^{- 1} and B, and a lower bound on the minimum eigenvalue q (A {star, open} B) for the Fan product of A and B are given. In addition, an upper bound on the spectral radius ρ (A {ring operator} 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.