`\[ \text{tr}\exp(A+B)\le \text{tr} \exp(A)\exp(B).\]`

Equality holds when $A$ and $B$ commutte; in fact, this is the only case equality occurs.
I first read about this inequality via a blog post by Lior Pachter of Caltech. He interpreted the inequality in the context of substitution matrices used in analyzing DNA mutations on a phylogenetic tree.

A thorough discussion can be found on Terence Tao's blog.

The inequality is itself interesting in the mathematical standpoint as it gives a qualitative comparision between two related expressions $\exp(A+B)$ and $\exp(A)\exp(B)$. One can ask for generalizations to three or higher number of matrices. Elliott H. Lieb found a three-matrix-version of the inequality in his paper *Convex trace functions and the Wigner-Yanase-Dyson conjecture* (The generalization is at the bottom of page 282.). I refer to the paper of Sutter, Berta, and Tomamichel and references therein for further discussion.