A Hardy–Hilbert-Type Inequality Involving Parameters Composed of a Pair of Weight Coefﬁcients with Their Sums

: In this paper, we establish a new Hardy–Hilbert-type inequality involving parameters composed of a pair of weight coefﬁcients with their sum. Our result is a uniﬁed generalization of some Hardy–Hilbert-type inequalities presented in earlier papers. Based on the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed, and the equivalent forms and the operator expressions are also considered. As applications, we illustrate how the inequality obtained can generate some new Hardy–Hilbert-type inequalities.

A sharpened version of inequality (1) was included in [1] by Theorem 323, as follows.
Motivated by the above-mentioned inequalities (2), (7) and (8), in this paper, we establish a new inequality that contains parameters composed of a pair of weight coefficients η 1 and η 2 with their sum η (η ∈ [0, 1 2 ]). The obtained inequality is a unified generalization of inequalities (2) and (7), as well as a more accurate version of inequalities (2) and (7). The main technical approaches include the construction of weight coefficients and the use of Hermite-Hadamard's inequality and the Euler-Maclaurin summation formula for estimation. Based on the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed. As applications, we deal with some equivalent forms, the operator expressions and some special cases for the inequalities obtained in the main result.
The rest of the paper is organized as follows: In Section 2, we provide some necessary notations, formulas and lemmas. In Sections 3 and 4, we state our main results, while some new inequalities with their equivalent forms and the operator expressions are provided. In Section 5, we end the paper with some concluding remarks and future directions of this study.

Preliminaries
Let us first state the following specified conditions (C1) that we will use in what follows. We suppose that the following is the case.
. We also assume a m , b n ≥ 0 such that the following is the case.
Hereinafter, the Euler-Maclaurin summation formula will be very helpful for us to deal with the estimations of integrals, which is stated as follows (cf. [4,5]).
Lemma 1. Define the following weight coefficient.
In the following, we prove inequality (10) by considering two cases.
In view of the decreasingness property of the series and by letting u = t−η 2 m−η 1 , we obtain the following: Hence, we obtain inequality (10). Case (ii). If λ 2 ∈ [1, 3 2 ] ∩ (0, λ), then by means of the Euler-Maclaurin summation formula, we have the following: 2 is a Bernoulli function of first order, and h(m) is indicated as follows.
Integrating by parts, the following is the case.
Furthermore, we obtain the following: where h i (i = 1, 2, 3) are indicated as follows.
On the other hand, we also have the following case: where H(m) is indicated as follows.
The following is, thus, obtained.
Hence, we obtain the following inequalities.
In view of the results obtained in Case (i), we obtain inequality (10). Lemma 1 is proved.

Lemma 2.
Under the assumptions described in (C1), we have the following more accurate Hardy-Hilbert's inequality.
By using the Hölder inequality (cf. [20]): where K(m, n), A m , B n ≥ 0, the following is the case.
If there exists a constant M ≤ B(λ 1 , λ 2 ) such that (14) is valid when we replace B(λ 1 , λ 2 ) by M, then by specifically performing a substitution of a m = a m and b n = b n in (14), we have the following.
By inequality (15) and the decreasingness property of series, we obtain the following case.
By using inequality (12) and setting the following: we find the following case.
Then, we obtain the following.
Now, letting ε → 0 + at both sides of the above inequality, in view of the continuity of the Beta function, we find B(λ 1 , λ 2 ) ≤ M. Hence, M = B(λ 1 , λ 2 ) is the best possible constant factor in (14). This completes the proof for Lemma 3. (11), we find the following case:

Remark 2. Setting
and the following is also the case.
Hence, it follows that B( λ 1 , λ 2 ) ∈ R + = (0, ∞). Note that for the following case: we have λ 1 , λ 2 ≤ 3 2 ; thus, we can rewrite inequality (14) in the following form. (11) is the best possible, then for the following case: (11) is the best possible, then in view of the assumption in (11), we have the following.

Lemma 4. If the constant factorB
By applying Hölder inequality, we have the following case: and then we obtain B( λ 1 , (17) retains the form of equality.
It is easily to observe that (17) keeps the form of equality if and only if there exist constants A and B such that they are not both zero and (cf. [20]) the following case holds.

Main Results
Our main results are stated in the following theorems. Theorem 1. Under the assumptions described in (C1), we have the following inequality that is equivalent to inequality (11).
Furthermore, if the constant factor in (11) is the best possible; thus, so is the constant factor in (18).
Proof. Suppose that (18) is valid. By utilizing the Hölder inequality, we have the following case.
Then, by (18), we obtain (11). On the other hand, assuming that (11) is valid, we set the following.
If the constant factor in (11) is the best possible, then so is the constant factor in (18). Otherwise, by (19), we would reach a contradiction that the constant factor in (11) is not the best possible. This completes the proof for Theorem 1.
Proof. (i) ⇒ (ii). By (i), in view of the continuity of the Beta function, we have the following case: can be expressible as a convergent single integral.
By assuming that a ∈ l p,φ and setting the following case: we can rewrite inequality (18) as follows: namely, c ∈ l p,ψ 1−p .

Definition 2.
Define a more accurate Hardy-Hilbert's operator T : l p,φ → l p,ψ 1−p as follows: For any a ∈ l p,φ , there exists a unique representation c ∈ l p,ψ 1−p . Define the formal inner product of Ta and b ∈ l q,ψ and the norm of T as follows. As a direct consequence of Theorems 1 and 2, we obtain the following case.

Conclusions
In this paper, by means of the construction of weight coefficients, the idea of introduced parameters, the techniques of real analysis and with the help of the Euler-Maclaurin summation formula, a more accurate extension of Hardy-Hilbert's inequality is established in Theorem 1, which contains parameters composed of a pair of weight coefficients with their sum. The equivalent conditions of the best possible constant factor related to several parameters are provided in Theorem 2. We also consider the equivalent forms, the operator expressions and some particular inequalities in Theorem 3 and Remark 3. The results provided in lemmas and theorems provide a significant supplement to the inequalities of the Hardy-Hilbert type. For further study, we may use these results and methods to establish new Hardy-Hilbert's inequalities involving partial sums, and this would enable us to confront the extensions of the results obtained in [21]. We will also investigate some new generalizations of Hardy-Hilbert's inequality by using the extended Euler-Maclaurin summation formula by using the techniques of analytical inequalities (cf. [22,23]).