A converse of Loewner-Heinz inequality and applications to operator means

Let f(t) be an operator monotone function. Then A⩽B implies f(A)⩽f(B), but the converse implication is not true. Let A♯B be the geometric mean of A,B⩾0. If A⩽B, then B−1♯A⩽I; the converse implication is not true either. We will show that if f(λB+I)−1♯f(λA+I)⩽I for all sufficiently small λ>0, then f(λA+I)⩽f(λB+I) and A⩽B. Moreover, we extend it to multi-variable matrices means.