A proof of
$$\lambda_n(A){\rm tr}\,B\leq {\rm tr}\,AB\leq\lambda_1(A){\rm tr}\,B$$
where $\lambda_n$ is the $n$-th largest eigenvalue of $A$, and $A$, $B$ are positive semidefinite, is given in D.L. Kleinman and M. Athans, <A HREF="http://ieeexplore.ieee.org/document/1098852/">"The design of suboptimal linear time-varying systems"</A> (1968). The inequality still holds for $A$ real symmetric and $B$ positive semidefinite, as proven in S.D. Wang, T.S. Kuo, and H.F. Hsu, "Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation" (1986).