Reverse inequalities involving two relative operator entropies and two relative entropies Original Research Article

We shall discuss relation among Tsallis relative operator entropy Tp(AmidB), the relative operator entropy image by J.I. Fujii-Kamei, the Tsallis relative entropy Dp(Ashort parallelB) by Furuichi–Yanagi–Kuriyama and the Umegaki relative entropy S(A, B). We show the following result: Let A and B be strictly positive definite matrices such that M1I greater-or-equal, slanted A greater-or-equal, slanted m1I > 0 and M2I greater-or-equal, slanted B greater-or-equal, slanted m2I > 0. Put image and p set membership, variant (0, 1]. Then the following inequalities hold:imagewhere K(p) is the generalized Kantorovich constant defined byimageand the first inequality is the reverse one of the second known inequalty, in particularimagewhere S(1) is the Specht ratio defined byimageand the first inequality is the reverse one of the second known inequalty.