An operator inequality and self-adjointness

Given bounded positive invertible operators A and B on a Hilbert space , it is shown that the inequality AXA−1 + B−1XB 2 X holds for all bounded operators X of rank 1 if and only if B=f(A) for some increasing function f satisfying a certain simple inequality, which in the case when the spectrum of A is connected implies that B is a scalar multiple of A. As an application some consequences of the Corach–Porta–Recht type inequality in operator ideals are studied.