Reverses of Young and Heinz inequalities for positive linear operators

Abstract

Abstract Let A, B be invertible positive operators on a Hilbert space H. We present some improved reverses of Young type inequalities, in particular, ( 1 − ν ) 2 ν ( A ∇ B ) + ( 1 − ν ) 2 ( 1 − ν ) H 2 ν ( A , B ) ≥ 2 ( 1 − ν ) ( A ♯ B ) $$ (1-\nu)^{2\nu}(A\nabla B)+(1-\nu)^{2(1-\nu)}H_{2\nu}(A,B) \geq2(1-\nu ) (A\sharp B) $$ and ( 1 − ν ) 2 ν H 2 ν ( A , B ) + ( 1 − ν ) 2 ( 1 − ν ) ( A ∇ B ) ≥ 2 ( 1 − ν ) ( A ♯ B ) , $$ (1-\nu)^{2\nu}H_{2\nu}(A,B)+(1-\nu)^{2(1-\nu)}(A\nabla B) \geq2(1-\nu ) (A\sharp B), $$ where 0 ≤ υ ≤ 1 2 $0\leq\upsilon\leq\frac{1}{2}$ . We also give some new inequalities involving the Heinz mean for the Hilbert-Schmidt norm.