Abstract
In this study, a multiparameter Hardy–Hilbert-type inequality for double series is established, which contains partial sums as the terms of one of the series. Based on the obtained inequality, we discuss the equivalent statements of the best possible constant factor related to several parameters. Moreover, we illustrate how the inequality obtained can generate some new Hardy–Hilbert-type inequalities.
1. Introduction
In [1], Krnić and Pečarić proposed an interesting result on the extension of Hardy–Hilbert inequality with two parameters, as follows:where , , . The constant factor in (1) is the best possible, which is represented by the beta function.
Obviously, for , inequality (1) reduces to the classical Hardy–Hilbert inequality ([2], Theorem 315):
In a special case, when , inequality (1) leads to a generalization of Hilbert inequality, i.e.,
Recently, by introducing more parameters, Yang et al. [3] established a further extension of inequality (1), as follows:
Inspired by the inequality (5) above, in this study, we construct and prove a new Hardy–Hilbert-type inequality by replacing the term of the series with in the right-hand side of inequality (5). Our method is mainly based on real analysis techniques and the applications of the Euler–Maclaurin summation formula and Abel’s partial summation formula. For details of various clever uses of these techniques, we refer the readers to [4–10].
The rest of the study is organized as follows. We first give some lemmas on the construction of weight function and several identities and inequalities related to the weight function. The results are then applied to derive a multiparameter Hardy–Hilbert-type inequality containing partial sums as the terms of one of the series. Finally, we illustrate the applications of the obtained inequality in discovering new Hardy–Hilbert-type inequalities.
2. Some Lemmas
Let us first state the following specified conditions (C1) that we will use in what follows. We suppose that(i)(C1) , , . Furthermore, for the partial sum is defined by , such that with
Lemma 1 (See [11]). (i)Let , with , and let and denote, respectively, the Bernoulli functions and the Bernoulli numbers of -order. Then,(ii)In particular, when in view of , we have(iii)When in view of , we have(iv)(See [11]). If , then we have the following Euler–Maclaurin summation formulas:
Lemma 2. For , we define the weight coefficient as follows:Then, the following inequalities hold:where .
Proof. For fixed , we define the function byBy virtue of (10), we haveNote that . Integration by parts, we findthen from , it follows thatBy (8), (9), (10), and (11), we obtainThus, we havewhereIt is easy to observe that , where the function is defined byTherefore, we deduce that, for ,thus, it follows that .
We find that for , andHence, we have , and then setting, it follows thatOn the other hand, by (10), we haveWe have obtained that and .
For , by (8), we find andHence, we havethen, we obtainwhere we set satisfyingTherefore, we obtain the required inequalities (13). This completes the proof of Lemma 2.
Lemma 3. For , we have the following Hardy–Hilbert-type inequality with the internal variables:
Proof. In the same way of proving inequality (13), we can prove the following inequalities for the weight coefficient :By Hӧlder’s inequality ([12]), we obtainFurthermore, by using (13) and (31), we get inequality (30). Lemma 3 is proved.
Remark 1. In particular, forin (30), replacing by , in view of (6), one has
Lemma 4. For , the following inequality holds:
Proof. In view of and applying Abel’s summation by parts formula, we findSince and the fact that for ,by (36), we haveHence, we derive the inequality (35). The proof of Lemma 4 is complete.
3. Main Results
Theorem 1. Under the assumptions described in (C1), we have the following inequality containing partial sums as the terms of series:In particular, for withwe have the following inequality:
Proof. By virtue of the fact thatfrom (35), we obtainThen, by (34), we derive inequality (39). Theorem 1 is proved.
Remark 2. Putting in (39), we have
Theorem 2. If , then the constant factorin (44) is the best possible. Moreover, ifand the constant factor in (44) is the best possible, then we have .
Proof. If , then we find ,and then inequality (44) reduces toFor any , we set . Then, we haveIf there exists a positive constant , (48) is valid when we replace by . Then, we haveBy (50) and the decreasing property of series, we obtainBy employing (31) (for ) and settingwe deduce thatThus, we haveFor , in view of the continuity of the beta function, we find . Hence, is the best possible constant factor of (48).
On the other hand, since , we findand .
By using Hӧlder’s inequality ([12]), we obtainIf the constant factor in (44) is the best possible, then by (48) (for ), we have the following inequality:namely, . Hence, (56) keeps the form of equality.
We observe that (56) keeps the form of equality if and only if there exist constants and , such that they are not all zero satisfying ([12]) a.e. in .
Assuming that , we have a. e. in , and then, , that is, . The proof of Theorem 2 is complete.
4. Some Applications
Theorem 3. Under the assumptions described in (C1), we have the following inequality which is equivalent to inequality (39):In particular, for withwe have the following inequality which is equivalent to (41):
Proof. Assuming that (58) is valid, by using Hӧlder’s inequality ([12]), we obtainThen, by (58), we have (39). On the other hand, assuming that (39) is valid, we setIf , then (58) is naturally valid; if , then it is impossible that makes (58) valid, namely, . Suppose that . By (39), we haveHence, (58) is valid. Therefore, inequality (58) is equivalent to inequality (39). Theorem 3 is proved.
Remark 3. Putting in (58), we get the following inequality which is equivalent to (44).
Theorem 4. If , then the constant factorin (64) is the best possible. Moreover, ifand the constant factor in (64) is the best possible, then we have .
Proof. If the constant factor in (64) is not the best possible, then by (60) (for ), we would reach a contradiction that the same constant factor in (44) is not the best possible. On the other hand, ifand the constant factor in (64) is the best possible, then, in view of the equivalency of (44) and (64) and , the same constant factor in (20) is the best possible. Furthermore, by Theorem 2, we have . This completes the proof of Theorem 4.
Remark 4. If we take in (41) and (59), respectively, then forwe have the following inequalities with the best possible constant factor :(i)Taking in (69) and (70), respectively, we obtain the following inequalities with the best possible constant factor :(ii)Taking in (69) and (70), respectively, we get the following inequalities with the best possible constant factor :In particular, if we take , then both (71) and (73) reduce toand both (72) and (74) reduce towhich is the equivalent form of inequality (75).
5. Conclusions
In this study, by using the weight coefficients, idea of introducing parameters, Euler–Maclaurin summation formula, and Abel’s partial summation formula, a multiparameter Hardy–Hilbert-type inequality is established, which contains partial sums as terms in one of the series. As applications, several equivalent statements of the best possible constant factor related to several parameters are provided. Moreover, we illustrate that our main results can generate more Hardy–Hilbert-type inequalities such as the inequalities (69)–(76) asserted by Remark 4.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
BY carried out the mathematical studies and drafted the manuscript. SW and JL participated in the design of the study and performed the numerical analysis. All authors contributed equally in the preparation of this article. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11801093), the Characteristic Innovation Project (Natural Science) of Guangdong Provincial Colleges and Universities (2017KTSCX133), the Characteristic Innovation Project of Guangdong Provincial Colleges and Universities (2020KTSCX088), and the Natural Science Foundation of Fujian Province of China (2020J01365).