An upper bound of the logarithmic mean is given by a convex conbination of the arithmetic mean and the geometric mean. In addition, a lower bound of the logarithmic mean is given by a geometric bridge of the arithmetic mean and the geometric mean. In this paper, we study the bounds of the logarithmic mean. We give operator inequalities and norm inequalities for the fundamental inequalities on the logarithmic mean. We give monotonicity of the parameter for the unitarily invariant norm of the Heron mean, and give its optimality as the upper bound of the unitarily invariant norm of the logarithmic mean. We study the ordering of the unitarily invariant norms for the Heron mean, the Heinz mean, the binomial mean and the Lehmer mean. Finally, we give a new mean inequality chain as an application of the point-wise inequality.
We give a refined Young inequality which generalizes the inequality by Zou-Jiang. We also show the upper bound for the logarithmic mean by the use of the weighted geometric mean and the weighted arithmetic mean. Furthermore, we show some inequalities among the weighted means. Based on the obtained essential scalar inequalities, we give some operator inequalities. In particular, we give a generalization of the result by Zou-Jiang, that is, we show the operator inequalities with the operator relative entropy with the weighted parameter. Finally, we give the further generalized inequalities by the Tsallis operator relative entropy.
We give a refined Young inequality which generalizes the inequality by Zou-Jiang. We also show the upper bound for the logarithmic mean by the use of the weighted geometric mean and the weighted arithmetic mean. Furthermore, we show some inequalities among the weighted means.
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