By similar arguments as for the proof of the golden-thompson inequality (see "Log majorization and complementary Golden-Thompson type inequalities" by T.Ando and F.Hiai) we can show that for all A,B symmetric positive definite we have
$$\|\log(A)+\log(B)\|\_{tr}\leq \|\log(A^{1/2}BA^{1/2})\|\_{tr},$$
where $\log$ is the matrix logarithm and $\|\cdot\|\_{tr}$ is the trace norm, i.e. $\|A\|\_{tr}=\sqrt{tr(AA^\*)}$. My conjecture is that for each $n\in \mathbb{N}$ there exists a constant $c\_n$ such that
$$\|\log(A)+\log(B)\|\_{tr}\geq c\_n\|\log(A^{1/2}BA^{1/2})\|\_{tr}$$
for all symmetric positive definite matrices $A,B$ of dimension $n$. However I have no idea how to prove it. The question is interesting because it would show that the two metrics $d\_1$, $d\_2$ defined by
$$d\_1(A,B)=\|\log(A)-\log(B)\| \text{ and } d\_2(A,B)=\|\log(A^{-1/2}BA^{-1/2})\|$$
on the space of positive definite Matrices are strongly equivalent.



Equivalent metrics on symmetric positive definite matrices
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.


