# A New Dual Hardy-Hilbert's Inequality with some Parameters and its Reverse

• Zhong, Wuyi
• Accepted : 2008.12.11
• Published : 2009.09.30
• 30 6

#### Abstract

By using the improved Euler-Maclaurin summation formula and estimating the weight coefficients in this paper, a new dual Hardy-Hilbert's inequality and its reverse form are obtained, which are all with two pairs of conjugate exponents (p, q); (r, s) and a independent parameter ${\lambda}$. In addition, some equivalent forms of the inequalities are considered. We also prove that the constant factors in the new inequalities are all the best possible. As a particular case of our results, we obtain the reverse form of a famous Hardy-Hilbert's inequality.

#### Keywords

Euler-Maclaurin summation formula;dual Hardy-Hilbert's inequality;reverse form;best constant factor;equivalent form

#### References

1. G. H. Hardy, J. E. Littiewood and G. Pariolya, Inequalities, London: Cambridge University Press, 1952.
2. D. S. Mintrinovic, J. E. Pecaric and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Boston: Kluwer Academic Publishers, 1991.
3. B. Yang, On a generalization of Hilbert's double series theorem, Mathematical Inequalities and Applications, 5(2)(2002), 197-204.
4. M. Gao and B. Yang, On extended Hilbert's inequality, Proceedings of the American Mathematical Society, 126(3)(1998), 751-759, https://doi.org/10.1090/S0002-9939-98-04444-X
5. I. Brnetic and J. Pecaric, Generalization of Inequality of Hardy-Hilbert's Type Inequalities, Mathematical Inequalities and Applications, 7(2)(2004), 217-225.
6. W. Zhong, A reverse Hilbert's type integral inequality, International Journal of Pure and Applied Mathematics, 36(3)(2007), 353-360.
7. W. Zhong and B. Yang, On the extended form on the reverse Hardy-Hilbert's integral inequalities, Journal of Southwest China Normal University(Natural Science), 29(4)(2007), 44-48.
8. W. Zhong and B. Yang, On a Multiple Hilbert-type Integral Inequality with the Symmetric Kernel, Journal of Inequalities and Applications, 2007, Article ID 27962, 1-17. https://doi.org/10.1155/2007/27962
9. C. Zhao and L. Debnath, Some new inverse type integral Hilbert integral inequalities, Journal Mathematical Analysis and Applications, 245(2001), 248-265.
10. B. Yang, A Dual Hardy-Hilbert's Type Inequality and Generalizations, Advances in Mathematics, 35(1)(2006), 102-108.
11. B. Yang and Y. Zhu, Inequalities on the Hurwitz Zeta-Function Restricted to the Axis of Positive Reals, Acta Sunyatseni Universitatis(Natu.Scie.), 36(3)(1997), 30-35.
12. J. Kuang, Applied inequalities, Jinan: Shangdong Science and Technology Press, 2004, 5-20.