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On a strengthened Hardy-Hilbert type inequality
Journal of Inequalities and Applications volume 2013, Article number: 511 (2013)
Abstract
We derive a strengthenment of a Hardy-Hilbert type inequality by using the Euler-Maclaurin expansion for the zeta function and estimating the weight function effectively. As applications, some particular results are presented.
MSC:26D15.
1 Introduction
Let , , , and . Then one [1] has
where the constant factor and pq are best possible. Inequality (1.1) is well known as Hardy-Hilbert’s inequality, and inequality (1.2) is named a Hardy-Hilbert type inequality. Both of them are important in analysis and applications [2]. In recent years, many results about generalizations of this type of inequality were established (see [3]). Under the same conditions as (1.1) and (1.2), some Hardy-Hilbert type inequalities, which are similar to (1.1) and (1.2), have been studied and generalized by some mathematicians.
By introducing a parameter, Yang gave a generalization of inequality (1.2) with the best constant factor as follows:
If , , , , such that and , then
where the constant factor is best possible.
Furthermore, by introducing a parameter and two pairs of conjugate exponents, Zhong gave a generalization of inequality (1.3) with the best constant factor as follows:
If , , , , , , such that and , then
where the constant factor is best possible.
Recently, in [4], Jiang and Hua established an improvement of inequality (1.3) as follows:
If , , , , , for , and , , then
where .
In this paper, by introducing a parameter and estimating the weight coefficient, we obtain a strengthenment of inequality (1.4) and generalize inequality (1.5). As applications, some particular results are presented.
2 Some preliminary results
First, we need the following formula of the Riemann-ζ function (see [5]):
where , , , , . The numbers , , , , … are Bernoulli numbers. In particular, ().
Since , the formula of the Riemann-ζ function (2.1) also holds for .
Lemma 2.1 Let , , , define the weight coefficients and as
Then we have
and
where .
Proof For , taking , in (2.1), we get
where .
Set , , and we can derive
where .
Thus we get
Combining (2.6) and (2.7), we have
In (2.6), let , by , we obtain
Therefore, for , , , we obtain
Applying the above inequality, we obtain (2.4). Similarly, we can prove (2.5). The lemma is proved. □
3 Main results
Theorem 3.1 Assume that , , , , , , , such that and , then
where . Inequality (3.1) is equivalent to (3.2). In particular, we have the following equivalent inequalities:
Proof From Hölder inequality (see [6]), we have
Hence, by (2.4), (2.5), inequality (3.1) is true.
Setting as
by using (3.1), we have
Hence, we obtain
By (3.1), both (3.5) and (3.6) take the form of strict inequality, and we have (3.2).
On the other hand, suppose that (3.2) is valid, from Hölder inequality, we find
Then, by using (3.2), we have (3.1). Hence, (3.2) and (3.1) are equivalent. The proof of Theorem 3.1 is completed. □
Since , by Theorem 3.1, we have the following.
Corollary 3.2 Assume that , , , , , , , such that and , then
where . Inequality (3.7) is equivalent to (3.8).
For , by using (3.1) and (3.2), we have the following.
Corollary 3.3 Assume that , , , , , such that and , then
where . Inequality (3.9) is equivalent to (3.10). In particular, we have the equivalent inequalities as follows.
For , , by using (3.1) and (3.2), we have the following.
Corollary 3.4 Assume that , , , , , such that and , then
where . Inequality (3.13) is equivalent to (3.14). In particular, we have the equivalent inequalities as follows.
For , , by using (3.1) and (3.2), we have the following.
Corollary 3.5 Assume that , , , , , such that and , then
where . Inequality (3.17) is equivalent to (3.18). In particular, we have the equivalent inequalities as follows.
Set , combining (3.1) and (3.2), we have the following.
Corollary 3.6 Assume that , ,, , , , such that and , then
In particular, we have the equivalent inequalities as follows.
Taking , in (3.23) and (3.24), we have
Remark 3.1 For and in Theorem 3.1, we get the results of [4].
References
Hardy GH, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge; 1952.
Mitrinovic DS, Pecaric JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston; 1991.
Yang BC, Rassias TM: On the way of weight coefficient and research for the Hilbert-type inequalities. Math. Inequal. Appl. 2003, 6(4):625–658.
Jiang W-D, Hua Y: On improvement of a Hilbert-type inequality. Comput. Math. Appl. 2010, 60(3):629–633. 10.1016/j.camwa.2010.05.009
Yang BC: The Norm of Operator and Hilbert-Type Inequality. Science Press, Beijing; 2008.
Kuang J: Applied Inequalities. Shandong Science Press, Jinan; 2003.
Acknowledgements
The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. The first author was supported by the scientific research foundation of National Natural Science Foundation (51109180), the National Science & Technology Supporting Plan from the Ministry of Science & Technology of P.R. China (2011BAD29B08), the ‘111’ Project from the Ministry of Education of P.R. China and the State Administration of Foreign Experts Affairs of P.R. China (B12007), Fundamental Research Funds for the Central Universities (Z109021310) and the scientific research foundation of National Natural Science Foundation (51279167).
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Chen, DY., Chen, GS., Song, T. et al. On a strengthened Hardy-Hilbert type inequality. J Inequal Appl 2013, 511 (2013). https://doi.org/10.1186/1029-242X-2013-511
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DOI: https://doi.org/10.1186/1029-242X-2013-511