Reverse inequality to Golden–Thompson type inequalities: Comparison of eA+BeA+B and eAeBeAeB

Let A and B be n-by-n Hermitian matrices. ThenTreAeB⩽S(α)TreA+Bwhere α   is the condition number of eAeA and S(t)S(t) is the Specht ratio of the reverse arithmetic-geometric mean inequality. It is a sharp reverse result to the Golden–Thompson inequality. This can be extended to each eigenvalue. Equivalently there exists a unitary V such thateA/2eBeA/2⩽S(α)VeA+BV∗.eA/2eBeA/2⩽S(α)VeA+BV∗.We also show that there exists a unitary W such thatWeA+BW∗⩽S(α)eA/2eBeA/2.WeA+BW∗⩽S(α)eA/2eBeA/2.