A Reverse Hardy–Hilbert’s Inequality Containing Multiple Parameters and One Partial Sum

: In this work, by introducing multiple parameters and utilizing the Euler–Maclaurin summation formula and Abel’s partial summation formula, we first establish a reverse Hardy– Hilbert’s inequality containing one partial sum as the terms of double series. Then, based on the newly proposed inequality, we characterize the equivalent conditions of the best possible constant factor associated with several parameters. At the end of the paper, we illustrate that more new in ‐ equalities can be generated from the special cases of the reverse Hardy–Hilbert’s inequality.


Introduction
is the best possible constant factor. Inequality (1) is known in the literature as Hardy-Hilbert's inequality (see [1] where the constant factor ) , ( given by the beta function is the best possible one. By introducing more parameters, Yang, Wu and Chen [3] established a further generalization of Hardy-Hilbert's inequality (1) as follows: where 1 where 1 Yang, Wu and Huang [6] established a reverse Hardy-Hilbert's inequality with one partial sum as the term of the double series, as follows: As a further study of the development methods of Hardy-Hilbert-type inequalities, some unconventional methods are adopted. For example, a half-discrete Hilbert-type inequality with the multiple upper limit function and the partial sums was provided by [7]. A reverse Hardy-Hilbert-type integral inequality involving one derivative function was published by [8]. Inequalities (4)- (6) and the work of [7,8] are meaningful extensions of (2) based on the Euler-Maclaurin summation formula, Abel's partial summation formula and the techniques of real analysis. Some applications of Hardy-Hilbert-type inequalities in the real analysis and operator theory can be found in the monograph [9]. In [10], Hong gave an equivalent condition between the best possible constant factor and the parameters in the extension of (4). Some other similar results are provided by [11][12][13].
Inspired by the work of [4][5][6][7][8][9][10], in this paper, we construct a reverse Hardy-Hilbert's inequality which contains one partial sum and some extra parameters inside the weight coefficients, the reverse Hardy-Hilbert's inequality has different structural forms by comparing with existing results mentioned above. Our method is mainly based on some skillful applications of the Euler-Maclaurin summation formula and Abel's partial summation formula. By means of the newly proposed inequality, we then discuss the equivalent conditions of the best possible constant factor associated with several parameters. As applications, we deal with some equivalent forms of the obtained inequality and illustrate how to derive more reverse inequalities of Hardy-Hilbert type from the current results.

Preliminaries
For convenience, let us first state the following conditions (C1) that would be used repeatedly in subsequent section: , it follows that: then we have the following Euler-Maclaurin summation formula: In the following, we divide two cases of to prove (12).

Main Results
, by using (15), it follows that:  Furthermore, by means of (13), we obtain (16). The proof of Theorem 1 is complete. □

Remark 1. For
], 3 , from (11) and (12) (18) (16) By (19) and the decreasing property of the series, we obtain: Putting 0    into the above inequality, by virtue of the continuity of the beta function, we obtain is the best possible constant factor in (17).
(ii) For the case of is the best possible factor of (17) (for This completes the proof of Theorem 2. □

Theorem 3. Under the assumption (C1), if the constant factor
If the constant factor q p k k (16) is the best possible, then by using (23), we have the following inequality:  . Theorem 3 is proved. □    (17) and (26), we obtain the following reverse inequalities with the best possible constant factor  :