Parametric Mulholland-type inequalities. (English) Zbl 1457.26016
Summary: By means 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 some parameters, the operator expressions and some particular examples are considered.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
Keywords:
weight function; Mulholland-type inequality; equivalent statement; parameter; operator expressionReferences:
| [1] | L. E. Azar,The connection between Hilbert and Hardy inequalities, Journal of Inequalities and Applications, 2013, 2013: 452. · Zbl 1297.26042 |
| [2] | V. Adiyasuren, T. Batbold and M. Krni´c,Hilbert-type inequalities involving differential operators, the best constants and applications, Math. Inequal. Appl., 2015, 18(1), 111-124. · Zbl 1307.26018 |
| [3] | G. H. Hardy, J. E. Littlewood and G. Po´olya,Inequalities, Cambridge University Press, Cambridge, 1934. · JFM 60.0169.01 |
| [4] | Z. X. Huang and B. C. Yang,On a half-discrete Hilbert-type inequality similar to Mulholland’s inequality, Journal of Inequalities and Applications, 2013, 2013: 290. · Zbl 1282.26030 |
| [5] | Y. Hong Y and Y. M. Wen,A necessary and Sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor, Annals Mathematica, 2016, 37A(3), 329-336. · Zbl 1374.26055 |
| [6] | Y. Hong,On the structure character of Hilbert’s type integral inequality with homogeneous kernel and applications, Journal of Jilin University (Science Edition), 2017, 55(2), 189-194. · Zbl 1389.26041 |
| [7] | Y. Hong, Q. L. Huang, B. C. Yang and J. Q. Liao,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, 2017: 316. · Zbl 1386.26025 |
| [8] | Y. Hong, B. He and B. C. Yang,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 Mathematics Inequalities, 2018, 12(3), 777-788. · Zbl 1403.26020 |
| [9] | Z. X. Huang and B. C. Yang,Equivalent property of a half-discrete Hilbert’s inequality with parameters, Journal of Inequalities and Applications, 2018, 2018: 333. · Zbl 1498.26049 |
| [10] | J. C. Kuang,Applied inequalities, Shangdong Science and Technology Press, Jinan, China, 2004. |
| [11] | J. C. Kuang,Real analysis and functional analysis (continuation) (second volume), Higher Education Press, Beijing, China, 2015. |
| [12] | M. Th. Rassias and B. C. Yang,On half-discrete Hilbert’s inequality, Applied Mathematics and Computation, 2013, 220, 75-93. · Zbl 1329.26041 |
| [13] | M. Th. Rassias and B. C. Yang,A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function, Applied Mathematics and Computation, 2013, 225, 263-277. · Zbl 1334.26056 |
| [14] | M. Th. Rassias, B. C. Yang,On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function, Applied Mathematics and Computation, 2013, 242, 800-813. · Zbl 1334.26057 |
| [15] | J. S. Xu,Hardy-Hilbert’s inequalities with two parameters, Advances in Mathematics, 2007, 36(2), 63-76. |
| [16] | Z. T. Xie, Z. Zeng and Y. F. Sun,A new Hilbert-type inequality with the homogeneous kernel of degree -2, Advances and Applications in Mathematical Sciences, 2013, 12(7), 391-401. · Zbl 1296.26107 |
| [17] | D. M. Xin,A Hilbert-type integral inequality with the homogeneous kernel of zero degree, Mathematical Theory and Applications, 2010, 30(2), 70-74. · Zbl 1503.26080 |
| [18] | D. M. Xin, B. C. Yang and A. Z. Wang,Equivalent property of a Hilbert-type integral inequality related to the beta function in the whole plane, Journal of Function Spaces, Volume 2018, Article ID2691816, 8 pages. · Zbl 1400.26066 |
| [19] | B. C. Yang,The norm of operator and Hilbert-type inequalities, Science Press, Beijing, China, 2009. |
| [20] | B. C. Yang,Hilbert-type integral inequalities, Bentham Science Publishers Ltd., The United Arab Emirates, 2009. · Zbl 1469.26004 |
| [21] | B. C. Yang,On the norm of an integral operator and applications, J. Math. Anal. Appl., 2006, 321, 182-192. · Zbl 1102.47036 |
| [22] | B. C. Yang,On the norm of a Hilbert’s type linear operator and applications, J. Math. Anal. Appl., 2007, 325, 529-541. · Zbl 1114.47010 |
| [23] | B. C. Yang and M. Krni´c,A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0, Journal of Mathematical Inequalities, 2012, 6(3), 401-417. · Zbl 1251.26014 |
| [24] | B. C. Yang and L. Debnath,Half-discrete Hilbert-type inequalities, World Scientific Publishing, Singapore, 2014. · Zbl 1296.26007 |
| [25] | B. C. Yang,On a more accurate multidimensional Hilbert-type inequality with parameters, Mathematical Inequalities and Applications, 2015, 18(2), 429-441. · Zbl 1326.26049 |
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