Squaring a reverse AM-GM inequality
Let A,B be positive operators on a Hilbert space with 0 < m ≤ A,B ≤ M . Then for every unital positive linear map Φ, Φ((A+B)/2) ≤ K(h)Φ(A♯B), and Φ((A+B)/2) ≤ K(h) (Φ(A)♯Φ(B)) , where A♯B is the geometric mean and K(h) = (h+1) 2 4h with h = M/m.