ON REVERSE WEIGHTED ARITHMETIC-GEOMETRIC MEAN INEQUALITIES FOR TWO POSITIVE OPERATORS

Abstract

Let A, B be positive operators on a Hilbert space with 0 < m ≤ A,B ≤ M.  Then for every positive unital linear map Φ,
     (A∇_µB)² ≤ \(\left[\frac{(M + m)^{2}}{4Mm}\right]^{2}\) (A#_µB)², 0 ≤ µ ≤ 1,
     Φ²(A∇_µB)² ≤ \(\left[\frac{(M + m)^{2}}{4Mm}\right]^{2}\) Φ²(A♯_µB)², 0 ≤ µ ≤ 1,
     Φ²(A∇_µB)² ≤ \(\left[\frac{(M + m)^{2}}{4Mm}\right]^{2}\) [Φ(A)♯_µΦ(B)]² , 0 ≤ µ ≤ 1.