Weighted arithmetic–geometric operator mean inequalities

In this paper, we refine and generalize some weighted arithmetic–geometric operator mean inequalities due to Lin (Stud. Math. 215:187–194, 2013 ) and Zhang (Banach J. Math. Anal. 9:166–172, 2015 ) as follows: Let A and B be positive operators. If 0 < m ≤ A ≤ m ′ < M ′ ≤ B ≤ M or 0 < m ≤ B ≤ m ′ < M ′ ≤ A ≤ M , then for a positive unital linear map Φ, Φ 2 ( A ∇ α B ) ≤ [ K ( h ) S ( h ′ r ) ] 2 Φ 2 ( A ♯ α B ) , Φ 2 ( A ∇ α B ) ≤ [ K ( h ) S ( h ′ r ) ] 2 [ Φ ( A ) ♯ α Φ ( B ) ] 2 , Φ 2 p ( A ∇ α B ) ≤ 1 16 [ K 2 ( h ) ( M 2 + m 2 ) 2 S 2 ( h ′ r ) M 2 m 2 ] p Φ 2 p ( A ♯ α B ) , Φ 2 p ( A ∇ α B ) ≤ 1 16 [ K 2 ( h ) ( M 2 + m 2 ) 2 S 2 ( h ′ r ) M 2 m 2 ] p [ Φ ( A ) ♯ α Φ ( B ) ] 2 p , where α ∈ [ 0 , 1 ] , K ( h ) = ( h + 1 ) 2 4 h , S ( h ′ ) = h ′ 1 h ′ − 1 e log h ′ 1 h ′ − 1 , h = M m , h ′ = M ′ m ′ , r = min { α , 1 − α } and p ≥ 2 .