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 −1 1 I H ≤ A ≤ h 1 I H and h −1 2 I H ≤ B ≤ h 2 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 (h 1 ,h 2) 4 p , 1 16 α 2p (h 1 , h 2) , α(h 1 , h 2) = (h 1 + h −1 1)σ(h 2 + h −1 2). Likewise, a p-th (p ≥ 2) power of the Diaz-Metcalf type inequality is also established.