Comments on conjectures of trace inequalities on a generalized Wigner-Yanase skew information

Furuichi, Yanagi and Kuriyama gave three conjectures of trace inequalities on the Wigner-Yanase-Dyson skew information and a generalized Wigner-Yanase skew information (Furuichi et al. (2009) [1]) and Yanagi found a counterexample showing that two of the three conjectures don't hold (Yanagi (2010) 16)). In this note, we show that the last conjecture does not hold in general. In addition, we show that in the case of 2 x 2 matrices the conjecture is true. (C) 2010 Elsevier Inc. All rights reserved.