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 ( λ B + I ) − 1 ♯ f ( λ A + I ) ⩽ I for all sufficiently small λ > 0 , then f ( λ A + I ) ⩽ f ( λ B + I ) and A ⩽ B . Moreover, we extend it to multi-variable matrices means.