On a New Extended Hardy–Hilbert’s Inequality with Parameters

: In this paper, by introducing parameters and weight functions, with the help of the Euler–Maclaurin summation formula, we establish the extension of Hardy–Hilbert’s inequality and its equivalent forms. The equivalent statements of the best possible constant factor related to several parameters are provided. The operator expressions and some particular cases are also discussed.

In 2016, by means of the technique of real analysis, Hong [20] considered some equivalent statements of the extensions of (1) with the best possible constant factor related to several parameters. For some similar works on the extensions of (3) and (4), we refer the reader to [21][22][23][24][25].
In a recent paper [26], Yang, Wu, and Wang gave a reverse half-discrete Hardy-Hilbert's inequality and its equivalent forms and dealt with their equivalent statements of the best possible constant factor related to several parameters.
Following the way of [20,26], in this paper, by the idea of introducing weight functions and parameters and using Euler-Maclaurin's summation formula, we give an extension of Hardy-Hilbert's inequality and its equivalent forms. The equivalent statements of the best possible constant factor related to several parameters are provided. We also discuss the operator expressions and some particular cases of these types of inequalities.
Proof. For any 0 < ε < pλ 1 , we set If the constant factor k λ (λ 1 ) in (13) is not the best possible, then there exists a positive constant M <k λ (λ 1 ), such that (13) is valid when replacing k λ (λ 1 ) by M. In particular, by substitution of a m = a m and b n = b n in (13), we have By (14) and the decreasingness property, we obtain By (11) and (12), (1)).
In view of the above results, we have For ε → 0 + , we find k λ (λ 1 ) ≤ M, which is a contradiction. Hence, M = k λ (λ 1 ) is the best possible constant factor of (13). Lemma 4 is proved.

If the constant factor
Recalling the Hölder's integral inequality with weight (cf. [27]): with equality holding if and only if there exist constants A and B (not all zero) such that A f p (x) = Bg q (x) a.e. in R + . By using Hölder's integral inequality, one has We observe that (16) keeps the form of equality if and only if there exist constants A and B (not all zero) such that Au λ−λ 2 −1 = Bu λ 1 −1 a.e. in R + .

Main Results
Theorem 1. Inequality (10) is equivalent to the following inequality: If the constant factor in (10) is the best possible, then so is the constant factor in (17). (17) is valid. By Hölder's inequality (cf. [27]), we have
If the constant factor in (10) is the best possible, then so is the constant factor in (17). Otherwise, by (18), we would reach a contradiction that the constant factor in (10) is not the best possible. The proof of Theorem 1 is complete.

Operator Expressions and Some Particular Cases
We define the functions: where from, Define the following real normed spaces: Assuming that a ∈ l p,φ , setting c = {c n } ∞ n=1 , c n := ∞ m=1 1 m λ + n λ a m , n ∈ N, we can rewrite (17) as follows namely, c ∈ l p,ψ 1−p .

Conclusions
We described the advancements compared to existing technologies and results like Inequalities (1), (2), and (4) in the introduction section. The first result obtained in the present paper, Inequality (10) asserted by Lemma 3, is the extended Hardy-Hilbert's inequality. The subsequent results are the equivalent forms of Inequality (10) and the equivalent statements of the best possible constant factor related to several parameters; these meaningful results are stated in Theorems 1 and 2, which have significant applications in the theory of inequalities. The operator expressions of the extended Hardy-Hilbert's inequality and its equivalent forms have wide applications in the theory of functional analysis. The idea and method presented in this paper can be spread for general use to investigate more inequalities involving infinite series or infinite integrals.