In , it is known that for any rotationally symmetric function , i.e. depends only on the radius , the following holds

By a simple calculation, it is easy to have

In other words, there holds

In , it is known that for any rotationally symmetric function , i.e. depends only on the radius , the following holds

By a simple calculation, it is easy to have

In other words, there holds

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Today we discuss the inferior limit of the product of two functions. Let us take the following simple question:

**Question**. Given a function with the following property

do we always have the following

It turns out that the statement should be hold since . Unfortunately, since the function blows up of order , the behavior of the product depends on the order of decay of the function . Let take the following counter-example.

We consider the function

Although the function is everywhere negative, there holds

Now it is clear that

By definition, one can easily prove that the statement of the question holds if we have