Young’s inequality and trace

Let φφ be a positive linear functional on Mn(C)Mn(C) and f,gf,g mutually conjugate in the sense of Young. In this note we show a necessary and sufficient condition for the inequality|φ(A∗B)|≦φf(|A|)+φg(|B|)(A,B∈Mn(C))in terms of φφ and ff. We also see that the inequalityφ(|A∗B|)≦φ(f(|A|))+φ(g(|B|))(A,B∈Mn(C))implies the trace property of φφ.