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

Abstract

Recently Tsallis relative operator entropy T p ( A∣ B) and Tsallis relative entropy D p ( A∥ B) are discussed by Furuichi–Yanagi–Kuriyama. We shall show two reverse inequalities involving Tsallis relative operator entropy T p ( A∣ B) via generalized Kantorovich constant K( p). As some applications of two reverse inequalities, we shall show two trace reverse inequalities involving −Tr[ T p ( A∣ B)] and D p ( 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 M 1 I ⩾ A ⩾ m 1 I > 0 and M 2 I ⩾ B ⩾ m 2 I > 0. Put m = m 2 M 1 , M = M 2 m 1 , h = M m = M 1 M 2 m 1 m 2 > 1 and p ∈ (0, 1]. Let Φ be normalized positive linear map on B( H). Then the following inequalities hold: (i) 1 - K ( p ) p Φ ( A ) ♯ p Φ ( B ) + Φ ( T p ( A | B ) ) ⩾ T p ( Φ ( A ) | Φ ( B ) ) ⩾ Φ ( T p ( A | B ) ) and (ii) F ( p ) Φ ( A ) + Φ ( T p ( A | B ) ) ⩾ T p ( Φ ( A ) | Φ ( B ) ) ⩾ Φ ( T p ( A | B ) ) , where K( p) is the generalized Kantorovich constant defined by K ( p ) = ( h p - h ) ( p - 1 ) ( h - 1 ) ( p - 1 ) p ( h p - 1 ) ( h p - h ) p and K( p) ∈ (0, 1] and F ( p ) = m p p h p - h h - 1 1 - K ( p ) 1 p - 1 ⩾ 0 . In addition, let A and B be strictly positive definite matrices, (iii) 1 - K ( p ) p ( Tr [ A ] ) 1 - p ( Tr [ B ] ) p + D p ( A ‖ B ) ⩾ - Tr [ T p ( A | B ) ] ⩾ D p ( A ‖ B ) and (iv) F ( p ) Tr [ A ] + D p ( A ‖ B ) ⩾ - Tr [ T p ( A | B ) ] ⩾ D p ( A ‖ B ) . In particular, both (iii) and (iv) yield the following known result: log S ( 1 ) Tr [ A ] + S ( A , B ) ⩾ - Tr [ S ^ ( A | B ) ] ⩾ S ( A , B ) , where S ( 1 ) = h 1 h - 1 e log h 1 h - 1 is said to be the Specht ratio and S(1) > 1.