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On a parametric Mulholland-type inequality and applications. (English) Zbl 1474.26153

Summary: In this paper, by the use of the weight functions, and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given. The equivalent statements of the best possible constant factor related to a few parameters are provided. As applications, the operator expressions and a few particular examples are considered.

MSC:

26D15 Inequalities for sums, series and integrals
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[1] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities, (1934), Cambridge University Press
[2] Yang, B. C., The Norm of Operator and Hilbert-Type Inequalities, (2009), Beijing, China: Science Press, Beijing, China
[3] Yang, B. C., Hilbert-Type Integral Inequalities, (2009), The United Arab Emirates: Bentham Science Publishers, The United Arab Emirates
[4] Krnić, M.; Pečarić, J., General hilbert’s and hardy’s inequalities, Mathematical Inequalities & Applications, 8, 1, 29-51, (2005) · Zbl 1079.26018
[5] Perić, I.; Vuković, P., Multiple Hilbert’s type inequalities with a homogeneous kernel, Banach Journal of Mathematical Analysis, 5, 2, 33-43, (2011) · Zbl 1223.26044
[6] Huang, Q. L., A new extension of Hardy-Hilbert-type inequality, Journal of Inequalities and Applications, 2015, article 397, (2015) · Zbl 1336.26033
[7] He, B.; Wang, Q., A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor, Journal of Mathematical Analysis and Applications, 431, 2, 889-902, (2015) · Zbl 1325.26050
[8] Xu, J. S., Hardy-Hilbert’s inequalities with two parameters, Advances in Mathematics, 36, 2, 63-76, (2007)
[9] Xie, Z. T.; Zeng, Z.; Sun, Y. F., A new Hilbert-type inequality with the homogeneous kernel of degree -2, Advances and Applications in Mathematical Sciences, 12, 7, 391-401, (2013) · Zbl 1296.26107
[10] Zhen, Z.; Gandhi, R. R. K.; Xie, Z. T., A new Hilbert-type inequality with the homogeneous kernel of degree -2 and with the integral, Bulletin of Mathematical Sciences and Applications, 3, 1, 11-20, (2014)
[11] Xin, D. M., A Hilbert-type integral inequality with the homogeneous kernel of zero degree, Mathematical Theory and Applications, 30, 2, 70-74, (2010)
[12] Azar, L. E., The connection between Hilbert and HARdy inequalities, Journal of Inequalities and Applications, 2013, article 452, (2013) · Zbl 1297.26042
[13] Adiyasuren, V.; Batbold, T.; Krnić, M., Hilbert-type inequalities involving differential operators, the best constants, and applications, Mathematical Inequalities & Applications, 18, 1, 111-124, (2015) · Zbl 1307.26018
[14] Rassias, M. T.; Yang, B., On half-discrete Hilbert’s inequality, Applied Mathematics and Computation, 220, 75-93, (2013) · Zbl 1329.26041
[15] Yang, B.; Krnić, M., A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0, Journal of Mathematical Inequalities, 6, 3, 401-417, (2012) · Zbl 1251.26014
[16] Rassias, M. T.; Yang, B., A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function, Applied Mathematics and Computation, 225, 263-277, (2013) · Zbl 1334.26056
[17] Rassias, M. T. H.; Yang, B. C., On a multidimensional half-discrete Hilbert - type inequality related to the hyperbolic cotangent function, Applied Mathematics and Computation, 242, 800-813, (2013) · Zbl 1334.26057
[18] Huang, Z.; Yang, B., On a half-discrete Hilbert-type inequality similar to Mulholland’s inequality, Journal of Inequalities and Applications, 2013, article 290, (2013) · Zbl 1282.26030
[19] Yang, B. C.; Debnath, L., Half-Discrete Hilbert-Type Inequalities, (2014), Singapore: World Scientific Publishing, Singapore
[20] Hong, Y.; Wen, Y., A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor, Annals Mathematica, 37A, 3, 329-336, (2016) · Zbl 1374.26055
[21] Hong, Y., On the structure character of Hilbert’s type integral inequality with homogeneous kernel and applications, Journal of Jilin University (Science Edition), 55, 2, 189-194, (2017)
[22] Hong, Y.; Huang, Q.; Yang, B.; Liao, J., The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications, Journal of Inequalities and Applications, 2017, article 316, (2017) · Zbl 1386.26025
[23] Xin, D.; Yang, B.; Wang, A., Equivalent property of a Hilbert-type integral inequality related to the beta function in the whole plane, Journal of Function Spaces, 2018, (2018) · Zbl 1400.26066
[24] Hong, Y.; He, B.; Yang, B. C., Necessary and sufficient conditions for the validity of Hilbert type integral inequalities with a class of quasi-homogeneous kernels and its application in operator theory, Journal of Mathematical Inequalities, 12, 3, 777-788, (2018) · Zbl 1403.26020
[25] Huang, Z. X.; Yang, B. C., Equivalent property of a half-discrete Hilbert’s inequality with parameters, Journal of Inequalities and Applications, 2018, article 333, (2018)
[26] Kuang, J. C., Applied Inequalities, (2004), Jinan, China: Shangdong Science and Technology Press, Jinan, China
[27] Kuang, J. C., Real Analysis and Functional Analysis (Continuation), 2, (2015), Beijing, China: Higher Education Press, Beijing, China
[28] Yang, B., On a more accurate multidimensional Hilbert-type inequality with parameters, Mathematical Inequalities & Applications, 18, 2, 429-441, (2015) · Zbl 1326.26049
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