Equivalent conditions of a multiple Hilbert-type integral inequality with the nonhomogeneous kernel. (English) Zbl 1490.26029
Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 116, No. 3, Paper No. 107, 21 p. (2022).
Summary: In this paper we prove equivalent conditions of a multiple Hilbert-type integral inequality with the general nonhomogeneous kernel and multi-parameters, using weight functions and employing methods of real analysis. We also deduce the related results for the case of homogeneous kernel as well as derive operator expressions with the norm. Finally, several examples with particular kernels are considered as well.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 47A07 | Forms (bilinear, sesquilinear, multilinear) |
Keywords:
multiple Hilbert-type integral inequality; kernel; weight function; norm; operator expressionReferences:
| [1] | Hardy, G. H., Littlewood, J. E., \(P \acute{o}\) lya, G.: Inequalities. Cambridge University Press, Cambridge, (1934) · Zbl 0010.10703 |
| [2] | Mitrinovi \(\acute{c} \), D. S., Pe \(\check{c}\) ari \(\acute{c},\) J. E., Fink, A. M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Acaremic Publishers, Boston, (1991) · Zbl 0744.26011 |
| [3] | Yang, BC, A survey of the study of Hilbert-type inequalities with parameters, Adv. Math., 28, 3, 257-268 (2009) · Zbl 1482.26047 |
| [4] | Burtseva, E.; Lundberg, S.; Persson, L-E; Samko, N., Multi-dimensional Hardy type inequalities in Holder spaces, J. Math. Inequ., 12, 3, 719-729 (2018) · Zbl 1410.46015 |
| [5] | Jaksetic, J.; Pecarc, J.; Kalamir, SK, Further improvement of an extension of Holder-type inequality, Math. Inequ. Appl., 22, 4, 1161-1175 (2019) · Zbl 1435.26022 |
| [6] | Batbold, T.; Azar, LE, A new form of Hilbert integral inequality, J. Math. Inequ., 12, 1, 379-390 (2018) · Zbl 1391.26047 |
| [7] | Fabelurin, OO; Oguntuase, JA; Persson, LE, Multidimensional Hardy-type inequality on time scales with variable exponents, J. Math. Inequ., 13, 3, 725-736 (2019) · Zbl 1425.26008 |
| [8] | Basci, Y.; Baleanu, D., Hardy-type inequalities within fractional derivatives without singular kernel, J. Inequ. Appl., 2018, 304 (2018) · Zbl 1498.26042 |
| [9] | Zhao, CJ; Cheung, WS, On Hilbert’s inequalities with alternating signs, J. Math. Inequ., 12, 1, 191-200 (2018) · Zbl 1391.26063 |
| [10] | You, MH; Guan, Y., On a Hilbert-type integral inequality with non-homogeneous kernel of mixed hyperbolic functions, J. Math. Inequ., 13, 4, 1197-1208 (2019) · Zbl 1435.26035 |
| [11] | Zhang, KW, A bilinear inequality, J. Math. Anal. Appl., 271, 288-296 (2002) · Zbl 1016.15015 |
| [12] | Yang, B. C.: On a extansion of Hilbert’s integral inequality with some parameters. Aust. J. Math. Anal. Appl. 1(1), Art.11 (2004) · Zbl 1086.26018 |
| [13] | Yang, BC, A new Hilbert’s type integral inequality, Soochow J. Math., 33, 4, 849-859 (2007) · Zbl 1137.26310 |
| [14] | Yang, BC, A Hilbert-type integral inequality with a non-homogeneous kernel, J. Xiamen Univ. (Nat. Sci.), 48, 2, 165-169 (2009) |
| [15] | Yang, BC, The Norm of Operator and Hilbert-Type Inequalities (2009), Beijin, China: Science Press, Beijin, China |
| [16] | Benyi, A., Oh, C.: Best constant for certain multilinear integral operator. J. Inequ. Appl. 2006, Art. ID. 28582, 1-12 · Zbl 1116.26011 |
| [17] | Hong, H., All-side generalization about Hardy-Hilbert integral inequalities, Acta Math. Sin., 44, 4, 619-625 (2001) · Zbl 1034.26017 |
| [18] | He, LP; Yu, J.; Gao, MZ, An extension of Hilbert’s integral inequality, J. Shaoguan Univ. (Nat. Sci.), 23, 3, 25-30 (2002) |
| [19] | Yang, BC, On a multiple Hardy-Hilbert’s integral inequality, Chin. Ann. Math., 24A, 6, 25-30 (2003) |
| [20] | Yang, BC; Rassias, ThM, On the way of weight coefficient and research for Hilbert-type inequalities, Math. Ineq. Appl., 6, 4, 625-658 (2003) · Zbl 1046.26012 |
| [21] | Yang, B. C., Brneti \(\acute{c},\) I., Krni \(\acute{c},\) M., Pe \(\check{c}\) ari \(\acute{c},\) J.: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Ineq. Appl. 8(2), 259-272 (2005) · Zbl 1078.26019 |
| [22] | Yang, BC, On the norm of an integral operator and applications, J. Math. Anal. Appl., 321, 182-192 (2006) · Zbl 1102.47036 |
| [23] | Yang, BC, On the norm of a self-adjoint operator and a new bilinear integral inequality, Acta Math. Sin. English Series, 23, 7, 1311-1316 (2007) · Zbl 1129.47011 |
| [24] | Milovanovic, GV; Rassias, MTh, Some properties of a hypergeometric function which appear in an approximation problem, J. Global Optim., 57, 1173-1192 (2013) · Zbl 1281.26010 |
| [25] | Rassias, MTh; Yang, BC, On half-discrete Hilbert’s inequality, Appl. Math. Comput., 220, 75-93 (2013) · Zbl 1329.26041 |
| [26] | Rassias, MTh; Yang, BC, A multidimensional half - discrete Hilbert - type inequality and the Riemann zeta function, Appl. Math. Comput., 225, 263-277 (2013) · Zbl 1334.26056 |
| [27] | Rassias, MTh; Yang, BC, On a multidimensional half - discrete Hilbert - type inequality related to the hyperbolic cotangent function, Appl. Math. Comput., 242, 800-813 (2013) · Zbl 1334.26057 |
| [28] | Milovanovic, GV; Rassias, MTh, Analytic Number Theory (2014), Springer, New York: Approximation Theory and Special Functions, Springer, New York · Zbl 1286.00055 |
| [29] | Krnić, M.; Vuković, P., Multidimensional Hilbert-type inequalities obtained via local fractional calculus, Acta Appl. Math., 169, 1, 667-680 (2020) · Zbl 1459.26036 · doi:10.1007/s10440-020-00317-x |
| [30] | Adiyasuren, V.; Batbold, T.; Krnić, M., Half-discrete Hilbert-type inequalities with mean operators, the best constants, and applications, Appl. Math. Comput., 231, 148-159 (2014) · Zbl 1410.26033 · doi:10.1016/j.amc.2014.01.011 |
| [31] | Adiyasuren, V.; Batbold, T.; Krnić, M., On several new Hilbert-type inequalities involving means operators, Acta Math. Sin. Engl. Ser., 29, 8, 1493-1514 (2013) · Zbl 1272.26009 |
| [32] | Brnetić, I.; Krnić, M., Multiple Hilbert and Hardy-Hilbert inequalities with non-conjugate parameters, Bull. Aust. Math. Soc., 71, 447-457 (2005) · Zbl 1079.26013 |
| [33] | Hong, Y.; Wen, YM, A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor, Ann. Math., 37A, 3, 329-336 (2016) · Zbl 1374.26055 |
| [34] | Hong, Y., On the structure character of Hilbert’s type integral inequality with homogeneous kernel and application, J. Jilin Univ. (Science Edition), 55, 2, 189-194 (2017) · Zbl 1389.26041 |
| [35] | Hong, Y.; Huang, QL; Yang, BC; Liao, JL, 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, J. Ineq. Appl., 2017, 316 (2017) · Zbl 1386.26025 |
| [36] | Xin, D. M., Yang, B. C., Wang, A. Z.: Equivalent property of a Hilbert-type integral inequality related to the beta function in the whole plane. J. Funct. Spaces Article ID2691816, 8 (2018) · Zbl 1400.26066 |
| [37] | Hong, Y.; He, B.; Yang, BC, 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, J. Math. Ineq., 12, 3, 777-788 (2018) · Zbl 1403.26020 |
| [38] | Huang, ZX; Yang, BC, Equivalent property of a half-discrete Hilbert’s inequality with parameters, J. Ineq. Appl., 2018, 333 (2018) · Zbl 1498.26049 |
| [39] | Yang, BC; Wu, SH; Wang, AZ, On a reverse half-discrete Hardy-Hilbert’s inequality with parameters, Mathematics, 7, 1054 (2019) |
| [40] | Wang, AZ; Yang, BC; Chen, Q., Equivalent properties of a reverse ’s half-discret Hilbert’s inequality, J. Ineq. Appl., 2019, 279 (2019) · Zbl 1499.26196 |
| [41] | Rassias, MTh; Yang, BC, On an equivalent property of a reverse Hilbert-type integral inequality related to the extended Hurwitz-zeta function, J. Math. Ineq., 13, 2, 315-334 (2019) · Zbl 1425.26015 |
| [42] | Huang, XS; Luo, RC; Yang, BC, On a new extended half-discrete Hilbert’s inequality involving partial sums, J. Ineq. Appl., 2020, 16 (2020) · Zbl 1503.26055 |
| [43] | Yang, BC; Huang, MF; Zhong, YR, Equivalent statements of a more accurate extended Mulholland’s inequality with a best possible constant factor, Math. Ineq. Appl., 23, 1, 231-244 (2020) · Zbl 1439.26049 |
| [44] | Yang, B. C., Wu, S. H., Wang, A. Z.: A new Hilbert-type inequality with positive homogeneous kernel and its equivalent form. Symmetric 12, 342 (2020). doi:10.3390/sym12030342 |
| [45] | Huang, ZX; Shi, YP; Yang, BC, On a reverse extended Hardy-Hilbert’s Inequality, J. Ineq. Appl., 2020, 68 (2020) · Zbl 1503.26056 |
| [46] | Rassias, MTh; Yang, BC; Raigorodskii, A., On Hardy-type integral inequality in the whole plane related to the extended Hurwitz-zeta fanction, J. Ineq. Appl., 2020, 94 (2020) · Zbl 1503.26075 |
| [47] | Liao, J. Q., Hong, Y., Yang, B. C.: Equivalent conditions of a Hilbert-type multiple integral inequality holding. Journal of Function Spaces, Volume 2020, Article ID 3050952, 6 · Zbl 1437.26031 |
| [48] | Wang, AZ; Yang, BC, Equivalent property of a more accurate half-discrete Hilbert’s inequality, J. Appl. Anal. Comput., 10, 3, 920-934 (2020) · Zbl 1457.26019 |
| [49] | Hong, Y.; Liao, JQ; Yang, BC; Chen, Q., A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applications, J. Inequ. Appl., 2020, 140 (2020) · Zbl 1503.26054 |
| [50] | Yang, B. C., Wu, S. H., Chen, Q.: A new extension of Hardy-Hilbert’s inequality containing kernel of double power functions. Mathematics 8, 339 (2020). doi:10.3390/math8060894 |
| [51] | Rassias, M. Th., Yang, B. C. , Raigorodskii A.: Equivalent properties of two kinds of Hardy-type integral inequalities. Symmetry 13, 1006 (2021). doi:10.3390/sym13061006 · Zbl 1488.26133 |
| [52] | Kuang, JC, Introduction to Real Analysis (1996), Chansha, China: Hunan Education Press, Chansha, China · Zbl 0856.26001 |
| [53] | Kuang, JC, Applied Inequalities (2004), Jinan, China: Shangdong Science Technic Press, Jinan, China |
| [54] | Huang, Q. L., Yang. B. C.: A multiple Hilbert-type inequality with a non-homogeneous kernel. J. Ineq. Appl. 2013:73 (2013). doi:10.1186/1029-242X-2013-73 · Zbl 1302.47006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
