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 $p$.