Theorem 1 For an arbitrary natural number k, the following inequality holds true:
where
Proof
Using Lemma 5 gives
where
Hence
Using Lemma 2 and taking in the right-hand side of inequality (13), respectively, we get
Adding up the above inequalities, we obtain
Theorem 1 is proved. □
Theorem 2 For an arbitrary natural number k, the following inequality holds true:
where
Proof
Utilizing Lemma 6 gives
where
Hence
Using Lemma 2 and taking in the left-hand side of inequality (13), respectively, we get
Adding up the above inequalities, we obtain
Theorem 2 is proved. □
Theorem 3 Let (), . Then
(14)
where is the best possible under the weight coefficient .
Proof
By Lemma 7, we have
Therefore, to prove inequality (14), it suffices to show that
(15)
Obviously, inequality (15) becomes an equality for . In what follows, we will assume that .
By Theorem 1 , we need only to prove that
Note that
it suffices to show
(16)
Substituting in (16), inequality (16) becomes
which is equivalent to the following inequality:
(17)
where
From the hypothesis , we have
Further, we have
Consequently, inequality (17) holds true, and inequality (14) is proved.
Let us now show that is the best possible under the weight coefficient .
Consider inequality (14) in a general form as
(18)
Putting in (18) yields
Thus the best possible value for in (18) should be .
This completes the proof of Theorem 3. □
Remark 1 From the definition of and in the same way as in [17], we can establish the following accurate estimates of :
(19)
Further, the approximation of can be derived as follows:
Remark 2 For , inequality (11) can be written as
(20)
It is easy to observe that
and
hence
This implies that inequality (14) is stronger than inequality (11).