{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:32:59Z","timestamp":1760149979949,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T00:00:00Z","timestamp":1696896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62166011","222068071"],"award-info":[{"award-number":["62166011","222068071"]}]},{"name":"Innovation Key Project of Guangxi Province","award":["62166011","222068071"],"award-info":[{"award-number":["62166011","222068071"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>By means of the weight functions, the idea of introduced parameters and the transfer formulas, two multidimensional Hilbert-type integral inequalities with the general nonhomogeneous kernel as H(||x||\u03b1\u03bb1||y||\u03b2\u03bb2)(\u03bb1,\u03bb2\u22600) are given, which are some extensions of the Hilbert-type integral inequalities in the two-dimensional case. Some equivalent conditions of the best value and several parameters related to the new inequalities are provided. Two corollaries regarding the kernel, represented as k\u03bb(||x||\u03b1\u03bb1,||y||\u03b2\u03bb2)(\u03bb1,\u03bb2\u22600), are given, and a few new inequalities for the particular parameters are obtained.<\/jats:p>","DOI":"10.3390\/axioms12100956","type":"journal-article","created":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T02:20:52Z","timestamp":1696990852000},"page":"956","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Equivalent Statements of Two Multidimensional Hilbert-Type Integral Inequalities with Parameters"],"prefix":"10.3390","volume":"12","author":[{"given":"Yiyuan","family":"Li","sequence":"first","affiliation":[{"name":"School of Art and Design, Guilin University of Electronic Technology, Guilin 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanru","family":"Zhong","sequence":"additional","affiliation":[{"name":"School of Computer Science and Information Security, Guilin University of Electronic Technology, Guilin 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bicheng","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Guangdong University of Education, Guangzhou 510303, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,10]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and Polya, G. 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Wuhan Univ."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/956\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:04:29Z","timestamp":1760130269000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/956"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,10]]},"references-count":18,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["axioms12100956"],"URL":"https:\/\/doi.org\/10.3390\/axioms12100956","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,10,10]]}}}