### 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 ofthe Diaz-Metcalf type inequality is also established.

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