Suppose with . Define
Show that is finite for all and .
Proof. We consider the equivalent form of as follows (this is nothing but )
We firstly consider the case when . From the identity
if , then which implies . Therefore
Since provided and for and , then we have
Thus, is bounded from above for all . For , consider . More precisely,
where . Note that
If , then . Thus, is bounded from above in which implies that is bounded from below in . Similar, we can prove that is bounded from above in .
Finally, from the above estimates, clearly is of class , and thus, so is .