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.