SOME OPERATOR alpha-GEOMETRIC MEAN INEQUALITIES

In this paper, we refine an operator alpha-geometric mean inequality as follows: let Phi 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 alpha is an element of [0; 1], (Phi(A)#(alpha)Phi (B))(2) <= (K(h)/K-2r(h'))(2) Phi(2) (A#B-alpha), where K(h) = (h+1)(2)/4h, K (h') = (h'+1)(2)/4h' h = M/m, h' = M'/m' and r = min {alpha, 1 - alpha}.