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<=D for two real numbers m< M and let Phi be a unital positive linear map on B(H). We prove the subadditive inequality f(Phi(B))+f(Phi(C)<= Phi(f(A))+Phi(f(D)) and apply it to obtain several inequalities such as the Jensen-Mercer operator inequality and the Petrovic operator inequality.