Recently, we introduced class A defined by an operator inequality,
and also the definition of class A is similar to that of paranormality
defined by a norm inequality.
As generalizations of class A and paranormality,
Fujii-Nakamoto introduced class F$(p,r,q)$ and $(p,r,q)$-paranormality
respectively.
These classes are related to $p$-hyponormality and log-hyponormality.
The author showed more precise inclusion relations
among the families of class F$(p,r,q)$ and $(p,r,q)$-paranormality
than the results by Fujii-Nakamoto,
and he also showed the results
on powers of class F$(p,r,q)$ operators.
But some of the results on class F$(p,r,q)$ require
the assumption of invertibility.
In this paper, we shall remove the assumption of invertibility
from the results on invertible class F$(p,r,q)$ operators.
Moreover we shall show that the families of class F$(p,r,\frac{p+r}{\delta+r})$
and $(p,r,\frac{p+r}{\delta+r})$-paranormality are proper on $p$.