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.