{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:54:21Z","timestamp":1760057661724,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T00:00:00Z","timestamp":1739923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Mentorship Program at Guangzhou Huashang College","award":["2024HSDS10","12471176","HKGSXJ2022-37","2024KYCXTD018"],"award-info":[{"award-number":["2024HSDS10","12471176","HKGSXJ2022-37","2024KYCXTD018"]}]},{"name":"National Natural Science Foundation of China","award":["2024HSDS10","12471176","HKGSXJ2022-37","2024KYCXTD018"],"award-info":[{"award-number":["2024HSDS10","12471176","HKGSXJ2022-37","2024KYCXTD018"]}]},{"name":"Science and Technology Plan Project of Guangzhou Haizhu District","award":["2024HSDS10","12471176","HKGSXJ2022-37","2024KYCXTD018"],"award-info":[{"award-number":["2024HSDS10","12471176","HKGSXJ2022-37","2024KYCXTD018"]}]},{"name":"Nonlinear Partial Differential Equations Research Team Project of Guangdong University of Education","award":["2024HSDS10","12471176","HKGSXJ2022-37","2024KYCXTD018"],"award-info":[{"award-number":["2024HSDS10","12471176","HKGSXJ2022-37","2024KYCXTD018"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Hilbert integral inequalities are beautiful inequalities with a symmetric structure, and have attracted much attention because of their important applications in the study of integral operators, and the Hilbert-type integral inequality involving variable upper limit integral functions is a generalized form of the Hilbert integral inequality. In this paper, we use a construction theorem of the Hilbert-type n-ple integral inequality with homogeneous kernels to discuss a high-dimensional Hilbert-type integral inequality involving multivariate variable upper limit integral functions, and obtain the sufficiently necessary conditions for constructing inequalities and the formulas for optimal constant factors, which improve and generalize the existing results.<\/jats:p>","DOI":"10.3390\/axioms14020145","type":"journal-article","created":{"date-parts":[[2025,2,19]],"date-time":"2025-02-19T08:36:46Z","timestamp":1739954206000},"page":"145","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Constructive Conditions for a High-Dimensional Hilbert-Type Integral Inequality Involving Multivariate Variable Upper Limit Integral Functions and Optimal Constant Factors"],"prefix":"10.3390","volume":"14","author":[{"given":"Qian","family":"Zhao","sequence":"first","affiliation":[{"name":"Artificial Intelligence College, Guangzhou Huashang College, Guangzhou 511300, China"}]},{"given":"Yong","family":"Hong","sequence":"additional","affiliation":[{"name":"Artificial Intelligence College, Guangzhou Huashang College, Guangzhou 511300, China"}]},{"given":"Bing","family":"He","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,19]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and Polya, G. (1934). Inequalities, Gambridge University Press."},{"key":"ref_2","first-page":"19","article-title":"Some new Hilbert type inequalities","volume":"46","author":"Salem","year":"2006","journal-title":"Kyungpook Math. J."},{"key":"ref_3","first-page":"75","article-title":"Note on Hardy-Riesz\u2019s extension of Hilbert\u2019s inequality","volume":"6","author":"Xu","year":"1991","journal-title":"Chin. Q. J. Math."},{"key":"ref_4","first-page":"83","article-title":"A note on Hilbert\u2019s integral inequalitoes","volume":"13","author":"Yang","year":"1998","journal-title":"Chin. Q. J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"33","DOI":"10.15352\/bjma\/1313363000","article-title":"Multiple Hilbert\u2019s type inequalities with a homogeneous kernel","volume":"5","author":"Peric","year":"2011","journal-title":"Banach J. Math. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"315","DOI":"10.5486\/PMD.2005.3100","article-title":"Hilbert\u2019s inequalities and their reverses","volume":"67","author":"Krnic","year":"2005","journal-title":"Publ. Math. Debr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.7153\/jmi-2019-13-85","article-title":"On a Hilbert-type integral inequality with non-homogeneous kernel of mixed hyperbolic functions","volume":"13","author":"You","year":"2019","journal-title":"J. Math. Inequal."},{"key":"ref_8","first-page":"559","article-title":"On a relation between Hardy-Hilbert\u2019s inequality and Mulholland\u2019s inequality","volume":"49","author":"Yang","year":"2006","journal-title":"Acta Math. Sin. (Chin. Ser.)"},{"key":"ref_9","first-page":"329","article-title":"A necessary and sufficient conditions of that Hilbert type series inequality with homogeneous kernel has the best constant factor","volume":"37A","author":"Hong","year":"2016","journal-title":"Chin. Ann. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1215\/20088752-2017-0031","article-title":"Equivalent properties of a Hilbert-type integral inequality with the best constant factor related the Hurwitz zeta function","volume":"9","author":"Rassias","year":"2018","journal-title":"Ann. Funct. Anal."},{"key":"ref_11","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_12","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1186\/s13660-019-2280-7","article-title":"On a new Hilbert-type integral inequality involving the upper limit functions","volume":"2020","author":"Mo","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1186\/s13660-021-02551-9","article-title":"On a multiple Hilbert-type integrality involving the upper limit functions","volume":"2021","author":"Zhong","year":"2021","journal-title":"J. Inequal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1186\/s13660-023-02931-3","article-title":"A Hardy-Hilbert-type integral inequality involving two multiple upper-limit functions","volume":"2023","author":"Luo","year":"2023","journal-title":"J. Inequal. Appl."},{"key":"ref_15","unstructured":"Hong, Y., and He, B. (2023). Theory and Applications of Hilbert-Type Inequalities, Science Press."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/145\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:37:52Z","timestamp":1760027872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/145"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,19]]},"references-count":15,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["axioms14020145"],"URL":"https:\/\/doi.org\/10.3390\/axioms14020145","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,19]]}}}