%0 Journal Article %T On estimating some distances involving operator entropies via Riemannian metric %J Khayyam Journal of Mathematics %I Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures) %Z 2423-4788 %A Chergui, Mohamed %A El Hilali, Abdenbi %A El Wahbi, Bouazza %D 2022 %\ 01/01/2022 %V 8 %N 1 %P 94-101 %! On estimating some distances involving operator entropies via Riemannian metric %K Relative operator entropy %K Parametric relative operator entropy %K Tsallis relative operator entropy %K Operator inequalities %R 10.22034/kjm.2021.260901.2082 %X In this paper, we focus on geometric properties for relative operator entropy and its extensions for positive definite matrices by considering Riemannian metric. In particular, we prove that the Tsallis relative entropy $T_p(A|B)$ lies inside the sphere centered at the geometric mean of $A$ and $B$ with the radius equal to the half of the Riemannian distance between $A$ and $B$. Some numerical examples are given in the aim to verify the validity of the reverse of some results. %U https://www.kjm-math.org/article_144163_d97fa77685815f38312bbbec3567c492.pdf