Comparison of operator mean geodesics

The space of positive invertible operators of a unital C-algebra has a natural structure of reductive homogenious manifold with a Finsler metric. Then pairs of points A and B can be joined by a natural geodesic $A natural_t B = A ^{; ; ; 1/2}; ; ; (A^{; ; ; -1/2}; ; ; B A^{; ; ; -1/2}; ; ; )^t A^{; ; ; 1/2}; ; ; $ for $t in [0, 1]$, where is the geometric mean in the sense of Kubo and Ando. In this paper, we consider estimates of the upper bounds for the difference between the geodesic and extended interpolational paths by terms of the spectra of positive operators. As applications, we investigate some properties of the velocity vectors for interpolational paths. Also, we obtain estimates of the upper bounds for $alpha$ -operator divergence as a noncommutative version of the $alpha$ -divergence in the information geometry.