A new family of weighted operator means including the weighted Heron, logarithmic and Heinz means (Research on structure of operators by order and related topics)

Abstract

As generalizations of the weighted arithmetic, geometric and harmonic means of two positive numbers or operators, the weighted power and Heron means are well known. On the weighted logarithmic mean, some rescarchers gave the definitions in their own way. Among them, we focus on the definition by Pal, Singh, Moslehian and Aujla based on the Hermite-Hadamard inequality for convex functions. In this report, firstly we propose the notion of a transpose symmetric path of weighted M-means for a symmetric operator mean M. Next, we give an example of the transpose symmetric path of weighted M-means including the weighted logarithmic mean by Pal et al., which newly produces the weighted Heinz mean. From this argument, we get some relations among the weighted Heron, logarithmic and Heinz means.