Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions
Abstract
1. Introduction and Preliminary Knowledge
2. Notation Conventions and Preliminary Lemmas
3. Main Results
4. Applications
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Hardy, G.H.; Littlewood, J.E.; Polya, G. Inequalities; Cambridge University Press: Cambridge, UK, 1934. [Google Scholar]
- Yang, B.C. On Hilbert’s integral inequality. J. Math. Anal. 1998, 220, 778–785. [Google Scholar]
- Kuang, J.C. On new extension of Hilbert’s integral inequality. J. Math. Anal. Appl. 1999, 235, 608–614. [Google Scholar]
- Krnic, M.; Pecaric, J. General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 2005, 8, 29–51. [Google Scholar] [CrossRef]
- Chen, Q.; Yang, B.C. Half-discrete Hardy-Hilbert’s inequality with two internal variables. J. Inequal. Appl. 2013, 2013, 485. [Google Scholar] [CrossRef]
- Krnic, M.; Pecaric, J. Extension of Hilbert’s inequality. J. Math. Anal. Appl. 2006, 324, 150–160. [Google Scholar] [CrossRef]
- Yang, B.C. The Norm of Operator and Hilbert-Type Inequalities; Science Press: Beijing, China, 2009. [Google Scholar]
- Xie, Z.T. A new half-discrete Hilbert’s inequality with the homogeneous kernel of degree −4 μ. J. Zhanjiang Norm. Univ. 2011, 32, 13–19. [Google Scholar]
- Hong, Y.; Wen, Y.M. A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor. Ann. Math. 2016, 37A, 329–336. [Google Scholar]
- Hong, Y. On the structure character of Hilbert’s type integral inequality with homogeneous kernel and application. J. Jilin Univ. (Sci. Ed.) 2017, 55, 189–194. [Google Scholar]
- Wang, A.; Yang, B.C. Equivalent statements of a Hilbert-type integral inequality with the extended Hurwitz Zeta function in the whole plane. J. Math. Inequal. 2020, 14, 1029–1054. [Google Scholar] [CrossRef]
- Rassias, M.T.; Yang, B.C. Equivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta function. Adv. Oper. Theory 2017, 2, 237–256. [Google Scholar]
- Rassias, M.T.; Yang, B.C. On a equivalent property of a reverse Hilbert-type integral inequality related to the extended Hurwitz-zeta function. J. Math. Inequal. 2019, 13, 315–334. [Google Scholar] [CrossRef]
- Adiyasuren, V.; Batbold, T.; Azar, L.E. A new discrete Hilbert-type inequality involving partial sums. J. Inequal. Appl. 2019, 2019, 127. [Google Scholar] [CrossRef]
- Huang, X.Y.; Luo, R.C.; Yang, B.C.; Huang, X.S. A new reverse Mulholland’s inequality with one partial sum in the kernel. J. Inequal. Appl. 2024, 2024, 9. [Google Scholar] [CrossRef]
- Peng, L.; Yang, B.C. A new extended Mulholland’s inequality involving one partial Sums. Open Math. 2024, 22, 20240039. [Google Scholar] [CrossRef]
- Yang, B.C.; Wu, S.H. An improved version of the parameterized Hardy-Hilbert inequality involving two partial sums. Mathematics 2025, 13, 1331. [Google Scholar] [CrossRef]
- Wang, A.Z.; Yang, B.C. A new half-discrete Hilbert -type inequality involving higher-order derivative function and partial sum. J. Jilin Univ. (Sci. Ed.) 2023, 61, 1296–1304. [Google Scholar]
- Huang, X.Y.; Yang, B.C. On a more accurate half-discrete Mulholland-type inequality involving one multiple upper limit function. J. Funct. Spaces 2021, 2021, 6970158. [Google Scholar] [CrossRef]
- Yang, B.C.; Wu, S.H. A weighted generalization of Hardy-Hilbert-type inequality involving two partial sums. Mathematics 2023, 11, 3212. [Google Scholar] [CrossRef]
- Wang, A.Z.; Hong, Y.; Yang, B.C. On a new half-discrete Hilbert-type inequality with the multiple upper limit function and the partial sums. J. Appl. Anal. Comput. 2022, 12, 814–830. [Google Scholar] [CrossRef] [PubMed]
- Hong, Y.; He, B. Theory and Applications of Hilbert-Type Inequalities; Science Press: Beijing, China, 2023; pp. 347–353. [Google Scholar]
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Hong, Y.; He, B.; Zhao, Q. Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions. Axioms 2025, 14, 755. https://doi.org/10.3390/axioms14100755
Hong Y, He B, Zhao Q. Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions. Axioms. 2025; 14(10):755. https://doi.org/10.3390/axioms14100755
Chicago/Turabian StyleHong, Yong, Bing He, and Qian Zhao. 2025. "Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions" Axioms 14, no. 10: 755. https://doi.org/10.3390/axioms14100755
APA StyleHong, Y., He, B., & Zhao, Q. (2025). Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions. Axioms, 14(10), 755. https://doi.org/10.3390/axioms14100755