Inequalities for the trace of matrix product

To obtain estimates of solutions of Lyapunov and Riccati equations which frequently occur in the stability analysis and optimal control design in linear control theory, many researchers have attempted to determine upper and lower bounds for the product of two matrices in terms of the trace of one matrix and the eigenvalues of the other. Baksalary and Puntanen claimed ("An inequality for the trace of matrix product", ibid., vol. 37, no. 2, p. 239-40, 1992) that they had obtained a better estimate for the trace of the product of two matrices. The purpose of this note is to point out that their main result is incorrect and a counterexample is presented.<>