Proof of Theorem 1.1.

Without loss of generality, we assume that . We have

Set , where . By Theorem B, we need to show

By Proposition 5.1 in [10], we have

Hence, Theorem 1.1 will be proved if we show the following two propositions.

Proposition 2.1.

One has

Proof.

Write

where

Since implies , we have

For , by Markov's inequality, we get

From (2.7) and (2.8), we can get

Note that , where . By Lemma 1.5, we can assume that

Set . As , by (2.10), we have

So, when ,

By (2.12), we have

Set , then (referred by [4]). We can get

Then,

So, we get

Therefore,

By Lemma 1.6, noting that , for ,

For , we have

Then, for , , we have

For , we decompose it into two parts,

It is easy to see that

So,

Now, we estimate , by (2.23),

For , we have

From (2.24) and (2.25), we can get

Finally, , and we will get

then

Hence, (2.4) can be referred from (2.9), (2.17), (2.20), (2.26), and (2.28).

Proposition 2.2.

One has

Proof.

Consider the following:

We first estimate , for , by Markov's inequality,

Hence,

Now, we estimate . Here, , so

We have

We estimate first. Similar to the proof of (2.16), we have

then

By Lemma 1.6, for , we have

For , we have

Next, turning to , it follows that

then

For , it follows that

Finally, , we have

From (2.38) to (2.42), we can get

(2.29) can be derived by (2.32), (2.36), and (2.43).

Proof of Theorem 1.2.

Without loss of generality, we set . It is easy to see that

So, we only prove the following two propositions:

The proof of (2.45) can be referred to [6], and the proof of (2.46) is similar to Propositions 2.1 and 2.2.