The author presents an upper bound for the spectral radius of the Hadamard product of two nonegative matrices \[ \rho(A\circ B)\leq \max_{1\leq i\leq n}\{2a_{ii}b_{ii}+ \rho(A)\rho(B)-a_{ii}\rho(B)-b_{ii}\rho(A)\} \] where \(A=(a_{ij})\) and \(B=(b_{ij})\) are nonnegative \(n\times n\) matrices, and a lower bound for the minimal eigenvalue of the Fan product of two \(M\)-matrices \[ \tau(A\star B)\geq \min_{1\leq i\leq n}\{a_{ii}\tau(B)+b_{ii}\tau(A)-\tau(A)\tau(B)\}, \] where \(A=(a_{ij})\) and \(B=(b_{ij})\) are \(n\times n\) \(M\)-matrices, sharper than the corresponding results of \textit{R. A. Horn} and \textit{C. R. Johnson} [Topics in matrix analysis. Cambridge etc.: Cambridge University Press. (1991; Zbl 0729.15001)].