An operator inequality and its consequences

Let f be a continuous convex function on an interval J, let A, B, C, D be self-adjoint operators acting on a Hilbert space with spectra contained in J such that A + D = B + C   and A⩽m⩽B,C⩽M⩽DA⩽m⩽B,C⩽M⩽D for two real numbers m