{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T11:31:52Z","timestamp":1768995112764,"version":"3.49.0"},"reference-count":24,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,19]],"date-time":"2024-12-19T00:00:00Z","timestamp":1734566400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangzhou Huashang College Featured Research Project","award":["2024HSTS08"],"award-info":[{"award-number":["2024HSTS08"]}]},{"name":"Guangzhou Huashang College Featured Research Project","award":["HKGSXJ2022-37"],"award-info":[{"award-number":["HKGSXJ2022-37"]}]},{"name":"Guangzhou Huashang College Featured Research Project","award":["2023KTSCX096"],"award-info":[{"award-number":["2023KTSCX096"]}]},{"name":"Guangzhou Huashang College Featured Research Project","award":["2021ZDJS055"],"award-info":[{"award-number":["2021ZDJS055"]}]},{"name":"Science and Technology Plan Project of Guangzhou Haizhu District","award":["2024HSTS08"],"award-info":[{"award-number":["2024HSTS08"]}]},{"name":"Science and Technology Plan Project of Guangzhou Haizhu District","award":["HKGSXJ2022-37"],"award-info":[{"award-number":["HKGSXJ2022-37"]}]},{"name":"Science and Technology Plan Project of Guangzhou Haizhu District","award":["2023KTSCX096"],"award-info":[{"award-number":["2023KTSCX096"]}]},{"name":"Science and Technology Plan Project of Guangzhou Haizhu District","award":["2021ZDJS055"],"award-info":[{"award-number":["2021ZDJS055"]}]},{"name":"Featured Innovation Project of the General University of Guangdong Province","award":["2024HSTS08"],"award-info":[{"award-number":["2024HSTS08"]}]},{"name":"Featured Innovation Project of the General University of Guangdong Province","award":["HKGSXJ2022-37"],"award-info":[{"award-number":["HKGSXJ2022-37"]}]},{"name":"Featured Innovation Project of the General University of Guangdong Province","award":["2023KTSCX096"],"award-info":[{"award-number":["2023KTSCX096"]}]},{"name":"Featured Innovation Project of the General University of Guangdong Province","award":["2021ZDJS055"],"award-info":[{"award-number":["2021ZDJS055"]}]},{"name":"Key Construction Discipline Scientific Research Ability Promotion Project of Guangdong Province","award":["2024HSTS08"],"award-info":[{"award-number":["2024HSTS08"]}]},{"name":"Key Construction Discipline Scientific Research Ability Promotion Project of Guangdong Province","award":["HKGSXJ2022-37"],"award-info":[{"award-number":["HKGSXJ2022-37"]}]},{"name":"Key Construction Discipline Scientific Research Ability Promotion Project of Guangdong Province","award":["2023KTSCX096"],"award-info":[{"award-number":["2023KTSCX096"]}]},{"name":"Key Construction Discipline Scientific Research Ability Promotion Project of Guangdong Province","award":["2021ZDJS055"],"award-info":[{"award-number":["2021ZDJS055"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Hilbert-type integral inequalities, which feature a symmetric structure, are a significant class of inequalities with broad applications in the study of operator theory. Hilbert-type integral inequalities involving variable upper limit integral functions are a generalized form of the classical Hilbert-type integral inequalities. In this paper, we employ the construction theorem of homogeneous kernel Hilbert-type integral inequalities and the properties of the Gamma function to discuss the conditions for constructing a Hilbert-type integral inequality involving an upper limit integral function and the best constant factor. We derive the necessary and sufficient conditions for constructing this inequality and a formula for calculating the best constant factor, thereby improving the existing results. Finally, as an application, we consider some special cases of parameters.<\/jats:p>","DOI":"10.3390\/sym16121682","type":"journal-article","created":{"date-parts":[[2024,12,19]],"date-time":"2024-12-19T06:50:44Z","timestamp":1734591044000},"page":"1682","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Condition for the Construction of a Hilbert-Type Integral Inequality Involving Upper Limit Functions"],"prefix":"10.3390","volume":"16","author":[{"given":"Yong","family":"Hong","sequence":"first","affiliation":[{"name":"Artificial Intelligence College, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lijuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Artificial Intelligence College, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huasong","family":"Xiao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Guangdong University of Education, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,19]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. 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