{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:52:04Z","timestamp":1760147524779,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T00:00:00Z","timestamp":1675900800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Natural Science Foundation","award":["61772140","2020KTSCX088","NSh-775.2022.1.1"],"award-info":[{"award-number":["61772140","2020KTSCX088","NSh-775.2022.1.1"]}]},{"name":"the Characteristic innovation project of Guangdong Provincial Colleges and universities in 2020","award":["61772140","2020KTSCX088","NSh-775.2022.1.1"],"award-info":[{"award-number":["61772140","2020KTSCX088","NSh-775.2022.1.1"]}]},{"name":"the grant supporting leading scientific schools of Russia","award":["61772140","2020KTSCX088","NSh-775.2022.1.1"],"award-info":[{"award-number":["61772140","2020KTSCX088","NSh-775.2022.1.1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Hardy-type integral inequalities play a prominent role in the study of analytic inequalities, which are essential in mathematical analysis and its various applications, such as in the study of symmetry and asymmetry phenomena. In this paper, employing methods of real analysis and using weight functions, we investigate some equivalent conditions of two kinds of reverse Hardy-type integral inequalities with a particular non-homogeneous kernel. A few equivalent conditions of two kinds of reverse Hardy-type integral inequalities with a particular homogeneous kernel are deduced in the form of applications.<\/jats:p>","DOI":"10.3390\/sym15020463","type":"journal-article","created":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T02:55:54Z","timestamp":1675911354000},"page":"463","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Equivalent Conditions of the Reverse Hardy-Type Integral Inequalities"],"prefix":"10.3390","volume":"15","author":[{"given":"Michael Th.","family":"Rassias","sequence":"first","affiliation":[{"name":"Department of Mathematics and Engineering Sciences, Hellenic Military Academy, 16673 Vari Attikis, Greece"},{"name":"Moscow Institute of Physics and Technology, Institutskiy per, d. 9, Dolgoprudny 141700, Russia"},{"name":"Institute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Princeton, NJ 08540, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bicheng","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrei","family":"Raigorodskii","sequence":"additional","affiliation":[{"name":"Moscow Institute of Physics and Technology, Institutskiy per, d. 9, Dolgoprudny 141700, Russia"},{"name":"The Department of Statistics and Random Processes of the Mechanics and Mathematics Faculty of Lomonosov, Moscow State University, Leninskie Gory, 1, Moscow 119991, Russia"},{"name":"Institute of Mathematics and Computer Science, Buryat State University, 24a Smolin St., Ulan-Ude 670000, Russia"},{"name":"Caucasus Mathematical Center, Adyghe State University, ul. Pervomayskaya, 208, Maykop 385000, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,9]]},"reference":[{"key":"ref_1","first-page":"45","article-title":"Note on a theorem of Hilbert concerning series of positive terms","volume":"23","author":"Hardy","year":"1925","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_2","first-page":"1","article-title":"Bernerkungen sur Theorie der beschrankten Billnearformen mit unendlich vielen Veranderlichen","volume":"140","author":"Schur","year":"1911","journal-title":"J. Math."},{"key":"ref_3","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. (1934). Inequalities, Cambridge University Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u0107, D.S., Pecaric, J.E., and Fink, A.M. (1991). 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