Reverse inequalities involving two relative operator entropies and two relative entropies

Abstract

We shall discuss relation among Tsallis relative operator entropy T p ( A∣ B), the relative operator entropy S ^ ( A | B ) by J.I. Fujii-Kamei, the Tsallis relative entropy D p ( A∥ B) 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 M 1 I ⩾ A ⩾ m 1 I > 0 and M 2 I ⩾ B ⩾ m 2 I > 0. Put h = M 1 M 2 m 1 m 2 > 1 and p ∈ (0, 1]. Then the following inequalities hold: 1 - K ( p ) p ( Tr [ A ] ) 1 - p ( Tr [ B ] ) p + D p ( A | | B ) ⩾ - Tr [ T p ( A | B ) ] ⩾ D 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 the first inequality is the reverse one of the second known inequalty, in particular log S ( 1 ) Tr [ A ] + S ( A , B ) ⩾ - Tr [ S ^ ( A | B ) ] ⩾ S ( A , B ) where S(1) is the Specht ratio defined by S ( 1 ) = h 1 h - 1 e log h 1 h - 1 and the first inequality is the reverse one of the second known inequalty. It is known that K( p) ∈ (0, 1] for p ∈ (0, 1] and S(1) > 1.