Some Operator Inequalities for Operator Means and Positive Linear Maps

Abstract

     In this note, some operator inequalities for operator means and positive linear maps are investigated. The conclusion based on operator means is presented as follows: Let Φ : B(H) → B(K) be a strictly positive unital linear map and h ⁻¹₁I _H ≤ A ≤ h ₁ I _H and h ⁻¹₂I _H ≤ B ≤h₂ I_H for positive real numbers h ₁ , h ₂ ≥ 1. Then for p > 0 and an arbitrary operator mean σ,

(Φ(A)σΦ(B)) ^p ≤ α _p Φ ^p (Aσ ^∗ B)

where α _p = max{( α ² (h₁ ,h ₂ )/4)^p,(1/16)α^2p (h ₁ ,h ₂ ) } , α(h ₁ ,h ₂ ) = (h ₁+ h ⁻¹₁)σ(h ₂ + h ⁻¹₂). Likewise, a p-th (p ≥ 2) power ofthe Diaz-Metcalf type inequality is also established.