More results on weighted means

Abstract

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.

Further properties of accretive matrices

“…To our knowledge, the bounds for the weighted logarithmic mean for the positive matrices case and/or scalar case have not been known yet; see [17,19] for example.…”