  Loewner 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 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 the
mixture of parameterized ordered means in terms of the Loewner order. We also
show the relation between two families of parameterized ordered means
associated with the power mean, monotonically interpolating given two
parameterized ordered means.
