Recently, we introduced class A as a new class of operators
includind $p$-hyponormal and log-hyponormal operators.
Class A is 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-Jung-S.H.Lee-M.Y.Lee-Nakamoto introduced class A$(p,r)$
and Yamazaki-Yanagida introduced absolute-$(p,r)$-paranormality.
Moreover, Fujii-Nakamoto introduced class F$(p,r,q)$ and $(p,r,q)$-paranormality
which are further generalizations of these classes.
In this paper, we shall show 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 we shall also show several results
on class F$(p,r,q)$ and $(p,r,q)$-paranormality.