SOME OPERATOR α-GEOMETRIC MEAN INEQUALITIES

Abstract

In this paper, we refine an operator α-geometric mean inequality as follows: let Φ be a positive unital linear map and 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 each α ∈ [0, 1],

     (Φ(A)♯_αΦ(B))² ≤ \(\left(\frac{(K(h)}{K^{2r}(h′)}\right)^{2}\) Φ² (A♯_αB),

where K(h) = \(\frac{(h+1)^{2}}{4h}\), K(h′) = \(\frac{(h′+1)^{2}}{4h′}\), h = \(\frac{M}{m}\), h′ = \(\frac{M′}{m′}\) and r = min {α, 1−α}.