{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:04:41Z","timestamp":1760234681222,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,4]],"date-time":"2021-06-04T00:00:00Z","timestamp":1622764800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100014857","name":"National Natural Science Foundation of China-Guangdong Joint Fund","doi-asserted-by":"publisher","award":["61772140"],"award-info":[{"award-number":["61772140"]}],"id":[{"id":"10.13039\/501100014857","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Guangdong Provincial Colleges and universities","award":["2020KTSCX088"],"award-info":[{"award-number":["2020KTSCX088"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.<\/jats:p>","DOI":"10.3390\/sym13061006","type":"journal-article","created":{"date-parts":[[2021,6,7]],"date-time":"2021-06-07T01:56:40Z","timestamp":1623031000000},"page":"1006","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities"],"prefix":"10.3390","volume":"13","author":[{"given":"Michael Th.","family":"Rassias","sequence":"first","affiliation":[{"name":"Institute of Mathematics, University of Zurich, CH-8057 Zurich, Switzerland"},{"name":"Moscow Institute of Physics and Technology, Institutskiy per, d. 9, 141700 Dolgoprudny, Russia"},{"name":"Program in Interdisciplinary Studies, Institute for Advanced Study, 1 Einstein Drive, 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, 141700 Dolgoprudny, Russia"},{"name":"Faculty of Bioengineering and Bioinformatics, Moscow State University, 119991 Moscow, Russia"},{"name":"Institute of Mathematics and Computer Science, Buryat State University, 670000 Ulan-Ude, Russia"},{"name":"Caucasus Mathematical Center, Adyghe State University, 385012 Maykop, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,4]]},"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. 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