Abstract
In this paper, we shall present some reverse arithmetic-geometric mean operator inequalities for unital positive linear maps. These inequalities improve some corresponding results due to Xue (J. Inequal. Appl. 2017:283, 2017).
Highlights
1 Introduction Let m, m, m1, m2, m3, M, M, M1, M2 and M3 be scalars, I be the identity operator and the other capital letters be used to represent general elements of the C∗-algebra B(H) of all bounded linear operators acting on a Hilbert space(H, ·, · )
An operator A is said to be positive if Ax, x ≥ 0 for all x ∈ H and we write it as A ≥ 0, it is said to be strictly positive if Ax, x > 0 for all x ∈ H \ {0} and we write it as A > 0
For 0 < m ≤ A, B ≤ M, Tominaga [2] proved that the following operator reverse AM-GM inequality holds: A + B ≤ S(h)A B, (1.1)
Summary
Let m, m , m1, m2, m3, M, M , M1, M2 and M3 be scalars, I be the identity operator and the other capital letters be used to represent general elements of the C∗-algebra B(H) of all bounded linear operators acting on a Hilbert space(H, ·, · ). The operator norm is denoted by ·. A linear map Φ is positive if Φ(A) ≥ 0 whenever A ≥ 0. For A, B > 0 the μ-weighted arithmetic mean and μ-weighted geometric mean of A and B are defined, respectively, by. A μB = A1/2 A–1/2BA–1/2 μA1/2, where μ ∈ [0, 1], when μ = 1/2, we write A∇B and A B for brevity for A∇1/2B and A 1/2B, respectively. For 0 < m ≤ A, B ≤ M, Tominaga [2] proved that the following operator reverse AM-GM inequality holds:
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