Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications

Recently Tsallis relative operator entropy Tp(A B) and Tsallis relative entropy Dp(A B) are discussed by Furuichi–Yanagi–Kuriyama. We shall show two reverse inequalities involving Tsallis relative operator entropy Tp(A B) via generalized Kantorovich constant K(p). As some applications of two reverse inequalities, we shall show two trace reverse inequalities involving −Tr[Tp(A B)] and Dp(A B) and also a known reverse trace inequality involving the relative operator entropy by Fujii–Kamei and the Umegaki relative entropy S(A, B) is shown as a simple corollary. We show the following result: Let A and B be strictly positive operators on a Hilbert space H such that M1 I A m1 I > 0 and M2 I B m2 I > 0. Put , , and p (0, 1]. Let Φ be normalized positive linear map on B(H). Then the following inequalities hold: and F(p)Φ(A)+Φ(Tp(AB)) Tp(Φ(A)Φ(B)) Φ(Tp(AB)),where K(p) is the generalized Kantorovich constant defined by and K(p) (0, 1] and . In addition, let A and B be strictly positive definite matrices, and F(p)Tr[A]+Dp(A B) -Tr[Tp(AB)] Dp(A B).In particular, both (iii) and (iv) yield the following known result: where is said to be the Specht ratio and S(1) > 1.