Check out:

[Reverse inequality to Golden–Thompson type inequalities: Comparison of $e^{A+B}$ and $e^Ae^B$
Jean-Christophe Bourin, Yuki Seo (2007)](https://www.sciencedirect.com/science/article/pii/S0024379507002091), Linear Algebra and its Applications
Volume 426, Issues 2–3, 15 October 2007, Pages 312-316.