Let \(A\) and \(B\) denote positive semidefinite Hermitian matrices, let \(\alpha\) and \(\beta\) denote real numbers, and let \(\circ\) denote the Hadamard (piecewise) product of matrices. The authors first fully describe the solution of the equation \(\text{tr}(AB)^\alpha=\text{tr}(A^\alpha B^\alpha)\). Then they give inequalities between traces and eigenvalues of \((A\circ B)^\alpha\) and \(A^\alpha\circ B^\alpha\), inequalities between eigenvalues of \(A^\alpha\circ B^\alpha\) and \(A^\beta\circ B^\beta\), and inequalities between eigenvalues of principal submatrices of \(A^\alpha\) and \(A^\beta\).