Convergence of logarithmic trace inequalities via generalized Lie-Trotter formulae

We shall extend logarithmic trace inequalities shown by Bebiano et al. [N. Bebiano, R. Lemos, J. da Providencia, Inequalities for quantum relative entropy, preprint] and also by Hiai and Petz [The Golden-Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993) 153-185], by applying log majorization equivalent to an order preserving operator inequality. We shall generalize the Lie-Trotter formulae, which extend the original Lie-Trotter formula, and the α-mean variant of the original Lie-Trotter formula in Hiai-Petz [Linear Algebra Appl. 181 (1993) 153-185]. By using this generalized Lie-Trotter formulae, we shall consider the convergence of certain logarithmic trace inequalities, as some extensions of Bebiano et al. [N. Bebiano, R. Lemos, J. da Providencia, Inequalities for quantum relative entropy, preprint] and Hiai-Petz [The Golden-Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993) 153-185].