Check out:
Reverse inequality to Golden–Thompson type inequalities: Comparison of eA+BReverse inequality to Golden–Thompson type inequalities: Comparison of $e^{A+B}$ and $e^Ae^B$ Jean-Christophe Bourin, Yuki Seo (2007), Linear Algebra and eAeBits Applications Jean-Christophe BourinVolume 426, Yuki Seob (2007)Issues 2–3, 15 October 2007, Pages 312-316.
