Inequalities and limits of weighted spectral geometric mean

We establish some new properties of spectral geometric mean. In particular, we prove a log majorization relation between (Bts/2A(1−t)sBts/2)1/s and the t-spectral mean A♮tB:=(A−1♯B)tA(A−1♯B)t two positive semidefinite matrices A B, where A♯B is mean, dominant one. The limit involving also studied. then extend all results in context symmetric spaces negative curvature.