  Quantum entropy and skew information play important roles in quantum
information science. They are defined by the trace of the positive operators so
that the trace inequalities often have important roles to develop the
mathematical theory in quantum information science. In this paper, we study
some properties for information quantities in quantum system through trace
inequalities. Especially, we give upper bounds and lower bounds of Tsallis
relative entropy, which is a one-parameter extension of the relative entropy in
quantum system. In addition, we compare the known bounds and the new bounds,
for both upper and lower bounds, respectively. We also give an inequality for
generalized skew information by introducing a generalized correlation measure.
