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