Suppose with . Define
.
Show that is finite for all and .
Proof. We consider the equivalent form of as follows (this is nothing but )
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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 .