A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order

: In this article, a new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order is obtained, by using the weight functions, the mid-value theorem, and the techniques of real analysis. A few equivalent statements of the best possible constant factor related to several parameters are considered. As applications, the equivalent forms and some particular inequalities are provided

In this article, following the way of Hong et al. [17], by using the weight functions, the mid-value theorem, and the techniques of real analysis, a new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order is obtained, which is a new idea to extend the results of Adiyasuren et al. [16] into the field of reverses.The equivalent statements of the best possible constant factor related to several parameters are considered.As applications, the equivalent forms and some particular inequalities are provided.

Some lemmas
In what follows, we suppose that < p 0 , and ∞ = ∞ v( ) .We also assume that and for ≥ a 0 Note.In view of the assumption, we observe that then we find Lemma 1. Define the following weight function: We have the following inequality: is decreasing.For fixed > x 0, define the following positive function ) By the assumption, g t x ( ) is strictly decreasing with respect to ∈ + t R .In view of the decreasingness property of series, we have the following inequality: and then, Then, inequality (6) follows.
The lemma is proved.□ Lemma 2. We have the following reverse Hilbert-type inequality: Proof.
The lemma is proved.□ Lemma 3.For > t 0 and ∈ ∪ m N 0 { }, we have the following expression: Proof.For = m 0, equation ( 11) is naturally valid.For ∈ m N, by the assumption and the Note, for and By substitution of = k m 1,…, in the above expression, we obtain equation (11).The lemma is proved.□ Lemma 4. For > t 0, we have the following inequality: . Proof.
) and setting ) and by Abel's summation formula and the mid-value theorem, we have Hence, we have equation (12).The lemma is proved.□
then the constant factor

)) ( ( ))
A new reverse half-discrete Hilbert-type inequality  5 in equation ( 13) is the best possible.On the other hand, if the same constant factor in equation ( 13) is the best possible, then for we have + = λ λ λ.
For any where we indicate that .
We obtain that for ) such that equation ( 14) is valid, when we replace + + B λ λ , by M , then in particular, we have By the decreasingness property of series, we have Replacing λ by + + λ m 1, and setting in equation ( 5), we have Hence, by equation ( 6), we obtain Based on the above results, we have ͠ For → + ε 0 , in view of the continuity of the beta function, we obtain A new reverse half-discrete Hilbert-type inequality  7 namely, ) ) is the best possible constant factor in equation ( 14).On the other hand, for ( ( ) in equation ( 14), we have By the reverse Hölder's inequality (cf.[18]), we find Since is the best possible constant factor in equation (13), by equation ( 16), we have . It follows that equation (17) keeps the form of equality.We observe that equation (17) keeps the form of equality if and only if there exist constants A and B (cf. [18]), such that they are not both zero and a.e.
then the constant factor in equation (20) is the best possible.On the other hand, if same constant factor in equation (20) is the best possible, then for we have + = λ λ λ.
Proof.If + = λ λ λ, then by Theorem 2, for = m 0, the constant factor in equation ( 18) is the best possible.The constant factor in equation ( 20) is still the best possible.Otherwise, by equation ( 22), we would reach a contradiction that the constant in equation ( 18) is not the best possible.On the other hand, if the same constant factor in equation ( 20) is the best possible, then, by the equivalency of equations ( 20) and (18), in view of = J I q (see the proof of Theorem 3), we can still show that the constant factor in equation ( 18) is the best possible.By the assumption and Theorem 2 (for = m 0), we have The theorem is proved.□

Corollary 1 .
The following equivalent inequalities with the nonhomogeneous kernel are valid: