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 .

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