Some new bounds for the Hadamard product and the Fan product of matrices

If $A$ and $B$ are nonnegative matrices, a sharp upper bound on the spectral radius $\rho(A\circ B)$ for the Hadamard product of two nonnegative matrices is given, and the minimum eigenvalue $\tau(A\star B)$ of the Fan product of two $M$-matrices $A$ and $B$ is discussed. In addition, we also give a sharp lower bound on $\tau(A\circ B^{-1})$ for the Hadamard product of $A$ and $B^{-1}$. Several examples, illustrating that the given bound is stronger than the existing bounds, are also given.
Comment: 15 pages, Submitted to LAA on 2012.10.14