Strong monotonicity of operator functions

Let A, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace M such that M is invariant for A and B, and A\(M) = B\(M) We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0 less than or equal to A less than or equal to B, and B - A is of finite rank, A(t) less than or equal to B-t for some t > 1 if and only if the null space of B - A is invariant for A.