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

On bounds of Tsallis relative entropy and an inequality for generalized skew information