Mean inequalities for sector matrices involving positive linear maps

Let S α ( 0 ≤ α < π 2 ) stand for the set of all complex sector matrices and σ 1 , σ 2 be two operator means satisfying σ 1 ≤ σ 2 . Except some other assertions, it is also shown that for A , B ∈ S α , ℜ ( A σ 1 B ) ≤ sec 2 α ℜ ( A σ 2 B ) and ℜ ( A σ 2 B ) - 1 ≤ sec 2 α ℜ ( A σ 1 B ) - 1 . In addition, if σ i ∗ ≤ σ i , for i = 1 , 2 and Φ is a unital positive linear map, then Φ ℜ ( A σ 1 B ) - 1 ≤ sec 2 α ℜ ( Φ ( A - 1 ) σ 2 Φ ( B - 1 ) ) .