Mixture and interpolation of the parameterized ordered means

Abstract

AbstractLoewner partial order plays a very important role in metric topology and operator inequality on the open convex cone of positive invertible operators. In this paper, we consider a family G = { G n } n ∈ N of the ordered means for positive invertible operators equipped with homogeneity and properties related to the Loewner partial order such as the monotonicity, joint concavity, and arithmetic-G-harmonic weighted mean inequalities. Similar to the resolvent average, we construct a parameterized ordered mean and compare two types of mixtures of parameterized ordered means in terms of the Loewner order. We also show a relation between two families of parameterized ordered means associated with the power mean monotonic interpolating given two parameterized ordered means.