A general double inequality related to operator means and positive linear maps

Abstract

Let A, B ∈ B(H ) be such that 0 < b 1 I ≤ A ≤ a 1 I and 0 < b 2 I ≤ B ≤ a 2 I for some scalars 0 < b i < a i , i = 1, 2 and Φ : B(H ) → B(K ) be a positive linear map. Weshow that for any operator mean σ with the representing function f , the double inequality(1−α)µ and # α (∇ α , resp.) is the weighted geometric (arithmetic, resp.) mean for α ∈ (0, 1).As applications, we present several generalized operator inequalities including Diaz-Metcalf and reverse Ando type inequalities. We also give some related inequalities involving Hadamard product and operator means.

Refinements of a reversed AM–GM operator inequality

“…Also the operator Kantorovich inequality says that [6]. In the following result we show some refinements of (2.8) and (2.9).…”

On $f$-connections of positive definite matrices

“…is the identity function s → s, s ∈ g(J Namely, it is easy to verify that the spectrum Sp (Z) ⊂ J, where Z = A −1/2 BA −1/2 and J = [m, M ] with m = b 2 a 1 and M = a 2 b 1 . By weighted arithmetic-geometric inequality (see [8])…”