Abstract
In this paper, we employ iteration on operator version of the famous Young inequality and obtain more arithmetic-geometric mean inequalities and the reverse versions for positive operators. Concretely, we obtain refined Young inequalities with the Kantorovich constant, the reverse ratio type and difference type inequalities for arithmetic-geometric operator mean under different conditions.
Highlights
1 Introduction Throughout this paper, A, B are both positive operators on a Hilbert space H, and Bh(H) is the semi-space of all bounded linear self-adjoint operators on H
We may assume that A and B are invertible without loss of generality, A∇μB = ( – μ)A + μB and A μB = A / A– / BA– / μA /, where ≤ μ ≤
These inequalities have recently been improved by Furuichi [ ] as follows
Summary
Throughout this paper, A, B are both positive operators on a Hilbert space H, and Bh(H) is the semi-space of all bounded linear self-adjoint operators on H. Notation B+(H) is written as the set of all positive operators in Bh(H). We may assume that A and B are invertible without loss of generality, A∇μB = ( – μ)A + μB and A μB = A / A– / BA– / μA / , where ≤ μ ≤. When μ = / we write A∇B and A B for brevity, respectively; see Kubo and Ando [ ]
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