{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T19:10:34Z","timestamp":1760037034190,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T00:00:00Z","timestamp":1759795200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangzhou Huashang College Research Team Project","award":["2021HSKT03"],"award-info":[{"award-number":["2021HSKT03"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>By using the construction theorem of semi-discrete Hilbert-type inequalities with quasi-homogeneous kernels and real analysis techniques, this paper establishes a semi-discrete Hilbert-type inequality involving partial sums and variable upper limit integral functions, obtains the necessary and sufficient condition for constructing such an inequality, and, under certain conditions, derives the computational expression of the best constant factor. Finally, we discuss the boundedness and operator norm of the corresponding operator using the obtained results.<\/jats:p>","DOI":"10.3390\/axioms14100755","type":"journal-article","created":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:46:56Z","timestamp":1759841216000},"page":"755","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions"],"prefix":"10.3390","volume":"14","author":[{"given":"Yong","family":"Hong","sequence":"first","affiliation":[{"name":"Artificial Intelligence College, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0907-8379","authenticated-orcid":false,"given":"Bing","family":"He","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qian","family":"Zhao","sequence":"additional","affiliation":[{"name":"Artificial Intelligence College, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,7]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and Polya, G. (1934). Inequalities, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1006\/jmaa.1997.5877","article-title":"On Hilbert\u2019s integral inequality","volume":"220","author":"Yang","year":"1998","journal-title":"J. Math. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1006\/jmaa.1999.6373","article-title":"On new extension of Hilbert\u2019s integral inequality","volume":"235","author":"Kuang","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","first-page":"29","article-title":"General Hilbert\u2019s and Hardy\u2019s inequalities","volume":"8","author":"Krnic","year":"2005","journal-title":"Math. Inequal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1186\/1029-242X-2013-485","article-title":"Half-discrete Hardy-Hilbert\u2019s inequality with two internal variables","volume":"2013","author":"Chen","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.jmaa.2005.11.069","article-title":"Extension of Hilbert\u2019s inequality","volume":"324","author":"Krnic","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Yang, B.C. (2009). The Norm of Operator and Hilbert-Type Inequalities, Science Press.","DOI":"10.2174\/97816080505501090101"},{"key":"ref_8","first-page":"13","article-title":"A new half-discrete Hilbert\u2019s inequality with the homogeneous kernel of degree \u22124 \u03bc","volume":"32","author":"Xie","year":"2011","journal-title":"J. Zhanjiang Norm. Univ."},{"key":"ref_9","first-page":"329","article-title":"A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor","volume":"37A","author":"Hong","year":"2016","journal-title":"Ann. Math."},{"key":"ref_10","first-page":"189","article-title":"On the structure character of Hilbert\u2019s type integral inequality with homogeneous kernel and application","volume":"55","author":"Hong","year":"2017","journal-title":"J. Jilin Univ. (Sci. Ed.)"},{"key":"ref_11","first-page":"1029","article-title":"Equivalent statements of a Hilbert-type integral inequality with the extended Hurwitz Zeta function in the whole plane","volume":"14","author":"Wang","year":"2020","journal-title":"J. Math. Inequal."},{"key":"ref_12","first-page":"237","article-title":"Equivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta function","volume":"2","author":"Rassias","year":"2017","journal-title":"Adv. Oper. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"315","DOI":"10.7153\/jmi-2019-13-23","article-title":"On a equivalent property of a reverse Hilbert-type integral inequality related to the extended Hurwitz-zeta function","volume":"13","author":"Rassias","year":"2019","journal-title":"J. Math. Inequal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1186\/s13660-019-2087-6","article-title":"A new discrete Hilbert-type inequality involving partial sums","volume":"2019","author":"Adiyasuren","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1186\/s13660-024-03080-x","article-title":"A new reverse Mulholland\u2019s inequality with one partial sum in the kernel","volume":"2024","author":"Huang","year":"2024","journal-title":"J. Inequal. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"20240039","DOI":"10.1515\/math-2024-0039","article-title":"A new extended Mulholland\u2019s inequality involving one partial Sums","volume":"22","author":"Peng","year":"2024","journal-title":"Open Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Yang, B.C., and Wu, S.H. (2025). An improved version of the parameterized Hardy-Hilbert inequality involving two partial sums. Mathematics, 13.","DOI":"10.3390\/math13081331"},{"key":"ref_18","first-page":"1296","article-title":"A new half-discrete Hilbert -type inequality involving higher-order derivative function and partial sum","volume":"61","author":"Wang","year":"2023","journal-title":"J. Jilin Univ. (Sci. Ed.)"},{"key":"ref_19","first-page":"6970158","article-title":"On a more accurate half-discrete Mulholland-type inequality involving one multiple upper limit function","volume":"2021","author":"Huang","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Yang, B.C., and Wu, S.H. (2023). A weighted generalization of Hardy-Hilbert-type inequality involving two partial sums. Mathematics, 11.","DOI":"10.3390\/math11143212"},{"key":"ref_21","first-page":"814","article-title":"On a new half-discrete Hilbert-type inequality with the multiple upper limit function and the partial sums","volume":"12","author":"Wang","year":"2022","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_22","unstructured":"Hong, Y., and He, B. (2023). 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