  We establish some new properties of spectral geometric mean. In particular,
we prove a log majorization relation between $\left(B^{ts/2}A^{(1-t)s}B^{ts/2}
\right)^{1/s}$ and the $t$-spectral mean $A\natural_t B :=(A^{-1}\sharp
B)^{t}A(A^{-1}\sharp B)^{t}$ of two positive semidefinite matrices $A$ and $B$,
where $A\sharp B$ is the geometric mean, and the $t$-spectral mean is the
dominant one. The limit involving $t$-spectral mean is also studied. We then
extend all the results in the context of symmetric spaces of negative
curvature.
