| Citation: |
Bicheng Yang, Aizhen Wang. A HALF-DISCRETE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY WITH THE GENERAL NONHOMOGENEOUS KERNEL AND A LOWER LIMIT FUNCTION[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1293-1311. doi: 10.11948/20260046
|
A HALF-DISCRETE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY WITH THE GENERAL NONHOMOGENEOUS KERNEL AND A LOWER LIMIT FUNCTION
-
Abstract
By means of the weight functions, the idea of introduced parameters and Hardy's integral inequality, a new extended half-discrete multidimensional Hilbert-type inequality with the general nonhomogeneous kernel as well as one lower limit function is obtained. The equivalent statements of the best value in the new inequality related to several parameters are considered. The equivalent forms, the operator expressions and some particular cases are deduced.
-
-
References
[1] V. Adiyasuren, T. Batbold and E. A. Azar, A new discrete Hilbert-type inequality involving partial sums, Journal of Inequalities and Applications, 2019, 2019, 127. doi: 10.1186/s13660-019-2087-6 [2] V. Adiyasuren, T. Batbold and M. Krnić, Hilbert–type inequalities involving differential operators, the best constants and applications, Math. Inequal. Appl., 2015, 18, 111–124. [3] G. AlNemer, A. I. Saied, M. Abd. Zakarya, H. A. El-Hamid, Q. Bazighifan and M. H. Rezk, Some new reverse Hilbert's inequalities on time scales, Symmetry, 2021, 13, 2431. doi: 10.3390/sym13122431 [4] L. E. Azar, The connection between Hilbert and Hardy inequalities, Journal of Inequalities and Applications, 2013, 2013, 452. doi: 10.1186/1029-242X-2013-452 [5] B. Batiha, N. Alshammari, F. Aldosari, F. Masood, O. Bazighifan, Asymptotic and osci llatory properties for even-order nonlinear neutral differential equations with damping term, Symmetry, 2025, 17(1), 87. doi: 10.3390/sym17010087 [6] B. Batiha, N. Alshammari, F. Aldosari, F. Masood and O. Bazighifan, Nonlinear neutral delay differential equations: Novel criteria for oscillation and asymptotic behavior, Mathematics, 2025, 13(1), 147. doi: 10.3390/math13010147 [7] O. Bazighifan, N. Alshammari, F. Aldosari and L. F. Iambor, Oscillation solutions for nonlinear second-order neutral differential equations, Mathematics, 2025, 13(2), 247. doi: 10.3390/math13020247 [8] Q. F. Brevig, The best constant in a Hilbert-type inequality, Expo. Math., 2024, 42, 125530. doi: 10.1016/j.exmath.2023.125530 [9] Q. Chen, B. He, Y. Hong and Z. Li, Equivalent parameter conditions for the validity of half-discrete Hilbert-type multiple integral inequality with generalized homogeneous kernel, J. Funct. Spaces, 2020, 2020, 7414861. [10] A. A. El-Deeb and J. Awrejcewicz, Novel fractional dynamic Hardy–Hilbert-type inequalities on time scales with applications, Mathematics, 2021, 9, 2964. doi: 10.3390/math9222964 [11] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934. [12] B. He, A multiple Hilbert-type discrete inequality with a new kernel and best possible constant factor, Journal of Mathematical Analysis and Applications, 2015, 431, 982–990. [13] B. He, Y. Hong and Z. Li, Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with non-homogeneous, Journal of Inequalities and Applications, 2021, 2021, 64. doi: 10.1186/s13660-021-02593-z [14] Y. Hong, On the structure character of Hilbert's type integral inequality with homogeneous kernel and applications, J. Jilin Univ. Sci. Ed., 2017, 55(2), 189–194. [15] 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. doi: 10.1186/s13660-017-1592-8 [16] Y. Hong and Y. M. Wen, A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor, Ann. Math., 2016, 37A(3), 329–336. [17] Y. Hong, Y. R. Zhong and B. C. Yang, On a more accurate half-discrete multidimensional Hilbert-type inequality involving one derivative function of m-order, Journal of Inequalities and Applications, 2023, 2023, 74. doi: 10.1186/s13660-023-02980-8 [18] Y. Hong, Y. R. Zhong and B. C. Yang, A more accurate halif-discrete multidimensional Hilbert-type inequality involving one multiple upper limit function, Journal of inequalities and Applications, 2023, 2023, 74. doi: 10.1186/s13660-023-02980-8 [19] Q. L. Huang, A new extension of Hardy-Hilbert-type inequality, Journal of Inequalities and Applications, 2015, 2015, 397. doi: 10.1186/s13660-015-0918-7 [20] X. Y. Huang and B. C. Yang, A novel multidimensional refinement of the half-discrete Hardy–Hilbert inequality with a parameterized kernel and a partial sum term, Axioms, 2026, 16, 69. [21] M. Krnić and J. Pečarić, Extension of Hilbert's inequality, J. Math. Anal. Appl., 2006, 324(1), 150–160. doi: 10.1016/j.jmaa.2005.11.069 [22] M. Krnić and J. Pečarić, General Hilbert's and Hardy's inequalities, Mathematical Inequalities & Applications, 2005, 8(1), 29–51. [23] J. C. Kuang, Applied Inequalities, Shandong Science and Technology Press, Jinan, 2004. [24] J. C. Kuang, Real and Functional Analysis (Continuation), Vol 2, Higher Education Press, Beijing, 2015. [25] R. C. Luo, B. C. Yang and L. P. He, A Hardy-Hilbert-type integral inequality involving two multiple upper-limit functions, Journal of Inequalities and Applications, 2023, 2023, 19. doi: 10.1186/s13660-023-02931-3 [26] F. Masood, B. Batiha and O. Bazighifan, Asymptotic and oscillatory characteristics of solutions of neutral differential equations, J. Math. Comput. Sci., 2025, 39, 418–436. doi: 10.22436/jmcs.039.04.02 [27] H. M. Mo and B. C. Yang, On a new Hilbert-type integral involving the upper limit functions, Journal of Inequalities and Applications, 2020, 2020, 5. doi: 10.1186/s13660-019-2280-7 [28] L. Peng and B. C. Yang, A new multidimensional Hilbert-type inequality with two internal variables involving one partial sum, Journal of Applied Analysis and Computation, 2026, 16(3), 1–21. [29] I. Perić and P. Vuković, Multiple Hilbert's type inequalities with a homogeneous kernel, Banach Journal of Mathematical Analysis, 2011, 5(2), 33–43. doi: 10.15352/bjma/1313363000 [30] P. Vaziri and H. Khodabakhshian, Cauchy–Schwarz type inequalities in 2-Hilbert A-modules and its reverses, Asian-Eur. J. Math., 2023, 16, 2350133. doi: 10.1142/S1793557123501334 [31] K. Venkatachalam, B. Batiha, B. Almarri, M. S. Kumar and O. Bazighifan, Existence and uniqueness results of the AB-Caputo type derivative in impulsive fractional differential equations, J. Math. Comput. Sci., 2025, 39(4), 407–417. doi: 10.22436/jmcs.039.04.01 [32] P. Vuković, Refinements of local fractional Hilbert-type inequalities, Ukr. Math. J., 2023, 74, 29–37. [33] 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. [34] D. M. Xin, A Hilbert-type integral inequality with the homogeneous kernel of zero degree, Mathematical Theory and Applications, 2020, 30(2), 70–74. [35] J. S. Xu, Hardy-Hilbert's inequalities with two parameters, Adv. Math., 2007, 36(2), 63–76. [36] B. C. Yang, The Norm of Operator and Hilbert-Yype Inequalities, Science Press, Beijing, 2009. [37] M. Zakarya, A. I. Saied, G. AlNemer and M. H. Rezk, A study on some new reverse Hilbert-type inequalities and its generalizations on time scales, J. Math., 2022, 2022, 6285367. doi: 10.1155/2022/6285367 [38] Z. Zeng, K. Raja Rama Gandhi and Z. T. Xie, A new Hilbert-type inequality with the homogeneous kernel of degree -2 and with the integral, Bulletin of Mathematical Sciences and Applications, 2014, 3(1), 11–20. -
-
Export File
Citation
Bicheng Yang, Aizhen Wang. A HALF-DISCRETE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY WITH THE GENERAL NONHOMOGENEOUS KERNEL AND A LOWER LIMIT FUNCTION[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1293-1311. doi: 10.11948/20260046
DownLoad: