The authors prove that if \(A\) and \(B\) are two \(n\)-by-\(n\) nonzero positive semidefinite matrices and \(\|\cdot\|\) is a unitarily invariant norm on matrices satisfying \(\|\text{diag}(1,0,\dots, 0)\|\geq 1\), then the inequalities \[ {\| AB\|\over\| A\|\,\| B\|}\leq {\| A+ B\|\over\| A\|+\| B\|}\quad\text{and}\quad {\| A\circ B\|\over \| A\|\,\| B\|}\leq {\| A+ B\|\over\| A\|+\| B\|} \] hold, where \(AB\) and \(A\circ B\) denote the usual product and the Hadamard (or Schur) product of \(A\) and \(B\). They also elaborate on this theme and derive other inequalities concerning the parallel sum of \(A\) and \(B\) and the product and Hadamard product of \(A^\alpha\) and \(B^\alpha\) for \(\alpha\geq 1\).