A trace inequality arising from quantum information theory

We introduce a skew information of Lieb's typeSf,g(A,X)=Trf(A)Xg(A)X-Trf(A)g(A)X2for selfadjoint matrices A, X. We give conditions for f and g so that Sf,g is positive or negative. As another important application, we settle the problem posed by Yanagi, Furuichi and Kuriyama from quantum information theory.