A Hardy-Hilbert-type inequality involving modified weight coefficients and partial sums

Xianyong Huang; Shanhe Wu; Bicheng Yang; Xianyong Huang; Shanhe Wu; Bicheng Yang

doi:10.3934/math.2022350

AIMS Mathematics

2022, Volume 7, Issue 4: 6294-6310. doi: 10.3934/math.2022350

Previous Article Next Article

Research article

A Hardy-Hilbert-type inequality involving modified weight coefficients and partial sums

1.
Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China
2.
Department of Mathematics, Longyan University, Longyan 364012, China
3.
Institute of Applied Mathematics, Longyan University, Longyan 364012, China

Received: 30 October 2021 Revised: 14 January 2022 Accepted: 16 January 2022 Published: 19 January 2022
MSC : 26D15, 26D10, 47A05

In this article, we construct proper weight coefficients and use them to establish a Hardy-Hilbert-type inequality involving one partial sum. Based on this inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed. We also consider the equivalent forms and the operator expressions of the obtained inequalities. At the end of the paper, we demonstrate that more new Hardy-Hilbert-type inequalities can be derived from the special cases of the present results.
- weight coefficient,
- Euler-Maclaurin summation formula,
- Hardy-Hilbert-type inequality,
- partial sums,
- applications
Citation: Xianyong Huang, Shanhe Wu, Bicheng Yang. A Hardy-Hilbert-type inequality involving modified weight coefficients and partial sums[J]. AIMS Mathematics, 2022, 7(4): 6294-6310. doi: 10.3934/math.2022350

Related Papers:

Abstract

In this article, we construct proper weight coefficients and use them to establish a Hardy-Hilbert-type inequality involving one partial sum. Based on this inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed. We also consider the equivalent forms and the operator expressions of the obtained inequalities. At the end of the paper, we demonstrate that more new Hardy-Hilbert-type inequalities can be derived from the special cases of the present results.

References

[1]	G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge: Cambridge University Press, 1934.
[2]	M. Krnić, J. Pečarić, Extension of Hilbert's inequality, J. Math. Anal. Appl., 324 (2006), 150-160. https://doi.org/10.1016/j.jmaa.2005.11.069 doi: 10.1016/j.jmaa.2005.11.069
[3]	B. Yang, On a generalization of Hilbert double series theorem, J. Nanjing Univ. Math. Biquarterly, 18 (2001), 145-152.
[4]	V. Adiyasuren, T. Batbold, L. E. Azar, A new discrete Hilbert-type inequality involving partial sums, J. Inequal. Appl., 2019 (2019), 127. https://doi.org/10.1186/s13660-019-2087-6 doi: 10.1186/s13660-019-2087-6
[5]	B. C. Yang, The norm of operator and Hilbert-type inequalities, Beijing: Science Press, 2009.
[6]	M. Krnić, J. Pečarić, General Hilbert's and Hardy's inequalities, Math. Inequal. Appl., 8 (2005), 29-51. http://dx.doi.org/10.7153/mia-08-04 doi: 10.7153/mia-08-04
[7]	I. Perić, P. Vuković, Multiple Hilbert's type inequalities with a homogeneous kernel, Banach J. Math. Anal., 5 (2011), 33-43. https://doi.org/10.15352/bjma/1313363000 doi: 10.15352/bjma/1313363000
[8]	Q. L. Huang, A new extension of Hardy-Hilbert-type inequality, J. Inequal. Appl., 2015 (2015), 397. https://doi.org/10.1186/s13660-015-0918-7 doi: 10.1186/s13660-015-0918-7
[9]	B. He, Q. Wang, A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor, J. Math. Anal. Appl., 431 (2015), 990-902. https://doi.org/10.1016/j.jmaa.2015.06.019 doi: 10.1016/j.jmaa.2015.06.019
[10]	J. S. Xu, Hardy-Hilbert's inequalities with two parameters, Adv. Math., 36 (2007), 63-76.
[11]	Z. T. Xie, Z. Zeng, Y. F. Sun, A new Hilbert-type inequality with the homogeneous kernel of degree-2, Adv. Appl. Math. Sci., 12 (2013), 391-401.
[12]	Z. Zhen, K. Raja Rama Gandhi, Z. T. Xie, A new Hilbert-type inequality with the homogeneous kernel of degree -2 and with the integral, Bull. Math. Sci. Appl., 2014, 11-20.
[13]	D. M. Xin, A Hilbert-type integral inequality with the homogeneous kernel of zero degree, Math. Theor. Appl., 30 (2010), 70-74.
[14]	L. E. Azar, The connection between Hilbert and Hardy inequalities, J. Inequal. Appl., 2013 (2013), 452. https://doi.org/10.1186/1029-242X-2013-452 doi: 10.1186/1029-242X-2013-452
[15]	V. Adiyasuren, T. Batbold, M. Krnić, Hilbert-type inequalities involving differential operators, the best constants and applications, Math. Inequal. Appl., 18 (2015), 111-124. http://dx.doi.org/10.7153/mia-18-07 doi: 10.7153/mia-18-07
[16]	G. Datt, M. Jain, N. Ohri, On weighted generalized composition operators on weighted Hardy spaces, Filomat, 34 (2020), 1689-1700. https://doi.org/10.2298/FIL2005689D doi: 10.2298/FIL2005689D
[17]	M. A. Ragusa, Parabolic Herz spaces and their applications, Appl. Math. Lett., 25 (2012), 1270-1273. https://doi.org/10.1016/j.aml.2011.11.022 doi: 10.1016/j.aml.2011.11.022
[18]	B. Yang, M. T. Rassias, A. Raigorodskii, On an extension of a Hardy-Hilbert-type inequality with multi-parameters, Mathematics, 9 (2021), 2432. https://doi.org/10.3390/math9192432 doi: 10.3390/math9192432
[19]	Y. Hong, Y. Wen, A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor, Chin. Ann. Math., 37A (2016), 329-336.
[20]	J. C. Kuang, Applied inequalities, Jinan: Shangdong Science and Technology Press, 2004.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)