A reverse Hardy–Hilbert-type integral inequality involving one derivative function

In this article, by using weight functions, the idea of introducing parameters, the reverse extended Hardy–Hilbert integral inequality and the techniques of real analysis, a reverse Hardy–Hilbert-type integral inequality involving one derivative function and the beta function is obtained. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and some particular inequalities are also presented.

In this paper, following [21,23], by the use of weight functions, the idea of introducing parameters, the reverse extension of (1) and the technique of real analysis, a reverse Hardy-Hilbert-type integral inequality with the kernel 1 (x+y) λ+1 (λ > 0) involving one derivative function and the beta function is given. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and a few particular inequalities are obtained.

Some lemmas
In what follows, we assume that 0 < p < 1, 1 is a non-negative measurable function in R + = (0, ∞), and g(y) is a non-negative increasing differentiable function unless at finite points in R + , with g( By the definition of the gamma function, for λ, x, y > 0, the following expression holds (cf. [29]): where the gamma function is defined by

Lemma 1
For t > 0, we have the following expression: Lemma 2 Define the following weight functions: We have the following expressions: Proof Setting u = t x , we find namely, (8) follows. In the same way, we have (9).
The lemma is proved.

Lemma 3
We have the following reverse Hardy-Hilbert integral inequality involving one derivative function: Proof By the reverse Hölder inequality (cf. [30]), we obtain If (12) keeps the form of an equality, then there exist constants A and B, such that they are not all zero, satisfying We assume that A = 0. For fixed a.e. y ∈ (0, ∞), we have

Integration in the above expression, since for any
Therefore, by (8) and (9), we have (11). The lemma is proved.

Theorem 1 We have the following reverse Hardy-Hilbert-type integral inequality involving one derivative function:
In particular, for λ 1 + λ 2 = λ (or a = 0), we reduce (13) to the following: where the constant factor 1 λ B(λ 1 , λ 2 ) is the best possible.
Proof By (14) (for λ i =λ i (i = 1, 2)), since is the best possible constant factor in (17), we have the following inequality: By the reverse Hölder inequality (cf. [30]), we obtain It follows that (18) keeps the form of an equality. We observe that (18) keeps the form of an equality if and only if there exist constants A and B, such that they are not all zero and Au λ-λ 2 -1 = Bu λ 1 -1 a.e. in R + (cf. [30]). Assuming that A = 0, it follows that We have a = λλ 1λ 2 = 0, namely, λ 1 + λ 2 = λ. The theorem is proved.
Proof (i) ⇒ (ii). In view of the assumption and the continuity of the beta function, we find . Then (18) keeps the form of an equality. By the proof of Theorem 2, we have λ 1 + λ 2 = λ.
Remark 2 For a = 0 in (11), we have We conform that the constant factor B(λ 1 , λ 2 ) in (19) is the best possible. Otherwise, we would reach a contradiction by (15) (for a = 0): the constant factor in (14) is not the best possible.

Equivalent form and some particular inequalities
Theorem 4 Inequality (13) is equivalent to the following reverse Hardy-Hilbert-type integral inequality involving one derivative function: In particular, for λ 1 + λ 2 = λ (or a = 0), we reduce (20) to the equivalent form of (14) as follows: where the constant factor 1 λ B(λ 1 , λ 2 ) is the best possible.
On the other hand, assuming that (13) is valid, we set (20) is naturally valid; if J = 0, then it is impossible to make (20) valid, namely J > 0. Suppose that 0 < J < ∞. By (13), we have namely, (20) follows, which is equivalent to (13). The constant factor 1 λ B(λ 1 , λ 2 ) is the best possible in (21). Otherwise, by (22) (for a = 0), we would reach a contradiction: that the constant factor in (14) is not the best possible.
The theorem is proved.
Replacing x by 1 x , and then replacing x λ-1 f ( 1 x ) by f (x) in (13) and (20), by calculation, we have the following.

Corollary 1
The following reverse Hardy-Hilbert-type integral inequalities with the nonhomogeneous kernel involving one derivative function are equivalent: Moreover, λ 1 + λ 2 = λ (or a = 0) if and only if the constant factor 1 (23) and (24) is the best possible.

Conclusions
In this paper, following [21,23], by the use of weight functions, the idea of introducing parameters, the reverse extension of (1) and the technique of real analysis, a reverse Hardy-Hilbert-type integral inequality with the kernel 1 (x+y) λ+1 (λ > 0) involving one derivative function and the beta function is given in Theorem 1. The equivalent statements of the best possible constant factor related to several parameters are considered in Theorem 3. The equivalent form, the cases of non-homogeneous kernel and a few particular inequalities are obtained in Theorem 4, Corollary 1 and Remark 3. The lemmas and theorems provide an extensive account of this type of inequalities.