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 S^(A|B) 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 m=m2M1, M=M2m1, h=Mm=M1M2m1m2>1 and p ∈ (0, 1]. Let Φ be normalized positive linear map on B(H). Then the following inequalities hold:equation(i)1-K(p)pΦ(A)♯pΦ(B)+Φ(Tp(A|B))⩾Tp(Φ(A)|Φ(B))⩾Φ(Tp(A|B))and equation(ii)F(p)Φ(A)+Φ(Tp(A|B))⩾Tp(Φ(A)|Φ(B))⩾Φ(Tp(A|B)),F(p)Φ(A)+Φ(Tp(A|B))⩾Tp(Φ(A)|Φ(B))⩾Φ(Tp(A|B)),where K(p) is the generalized Kantorovich constant defined byK(p)=(hp-h)(p-1)(h-1)(p-1)p(hp-1)(hp-h)pand K(p) ∈ (0, 1] and F(p)=mpphp-hh-11-K(p)1p-1⩾0. In addition, let A and B be strictly positive definite matrices, equation(iii)1-K(p)p(Tr[A])1-p(Tr[B])p+Dp(A‖B)⩾-Tr[Tp(A|B)]⩾Dp(A‖B)and equation(iv)F(p)Tr[A]+Dp(A‖B)⩾-Tr[Tp(A|B)]⩾Dp(A‖B).F(p)Tr[A]+Dp(A‖B)⩾-Tr[Tp(A|B)]⩾Dp(A‖B).In particular, both  and  yield the following known result:logS(1)Tr[A]+S(A,B)⩾-Tr[S^(A|B)]⩾S(A,B),where S(1)=h1h-1elogh1h-1 is said to be the Specht ratio and S(1) > 1.