Monotonicity of order preserving operator functions

We discuss monotonicity of order preserving operator functions and related order preserving operator inequalities. Let A >= B >= 0 with A > 0, t is an element of [0, 1] and p >= 1. Let F(lambda, mu) = A(-lambda/2){A(lambda/2) (A(-1/2) B-p A(-1/2))(mu) A(lambda/2)}(1-t+lambda/(p-1)mu+lambda) A(-lambda/2). We show that: (i) F(r, w) >= F(r, 1) >= F(r, s) >= F(r, s') for any s' >= s >= 1, r >= t and 1-t/p-t <= w <= 1, (ii) F(q, s) >= F(t, s) >= F(r, s) >= F(r', s) for any r' >= r >= t, s >= 1 and t-1 <= q <= t. These imply the following recent inequality due to Kamei A(t) #(1-t/p-t) B-p >= A(1/2) F(r, s)A(1/2) for r >= t and s >= 1. (c) 2007 Elseiver Inc. All rights reserved.