Path of quasi-means as a geodesic

As a generalization of the Hiai–Petz geometries, we discuss two types of them where the geodesics are the quasi-arithmetic means and the quasi-geometric means respectively. Each derivative of such a geodesic might determine a new relative operator entropy. Also in these cases, the Finsler metric can be induced by each unitarily invariant norm. If the norm is strictly convex, then the geodesic is the shortest. We also give examples of the shortest paths which are not the geodesics when the Finsler metrics are induced by the Ky Fan k-norms.