A generalized skew information and uncertainty relation

A generalized skew information is defined and a generalized uncertainty relation is established with the help of a trace inequality which was recently proven by Fujii. In addition, we prove the trace inequality conjectured by Luo and Zhang. Finally, we point out that Theorem 1 in S. Luo and Q. Zhang, IEEE Trans. Inf. Theory, vol. 50, pp. 1778-1782, no. 8, Aug. 2004 is incorrect in general, by giving a simple counter-example.