A proof of
$$\lambda_n(B)\,{\rm tr}\,A\leq {\rm tr}\,(AB)\leq\lambda_1(B)\,{\rm tr}\,A$$
where $\lambda_n$ is the $n$-th largest eigenvalue of $B$ <sub>so $||B||=\lambda_1(B)$</sub> 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 $B$ real symmetric and $A$ positive semidefinite, as proven in S.D. Wang, T.S. Kuo, and H.F. Hsu, <A HREF="http://ieeexplore.ieee.org/document/1104370/">"Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation"</A> (1986). The restriction of positive semidefinite $A$ cannot be relaxed.