{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T19:35:52Z","timestamp":1775590552031,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T00:00:00Z","timestamp":1612137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we derive some new fractional extensions of Hardy\u2019s type inequalities. The corresponding reverse relations are also obtained by using the conformable fractional calculus from which the classical integral inequalities are deduced as special cases at \u03b1=1.<\/jats:p>","DOI":"10.3390\/sym13020242","type":"journal-article","created":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T05:16:28Z","timestamp":1612156588000},"page":"242","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Generalizations of Hardy\u2019s Type Inequalities via Conformable Calculus"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2222-7973","authenticated-orcid":false,"given":"Ghada","family":"AlNemer","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9752-4035","authenticated-orcid":false,"given":"Mohammed","family":"Kenawy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt"},{"name":"ASRT, 101 Kasr Al-Aini ST, Academy of Scientific Research &amp; Technology, Cairo 11334, Egypt"}]},{"given":"Mohammed","family":"Zakarya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, Universit\u00e0 Telematica Internazionale Uninettuno, 00186 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6782-7908","authenticated-orcid":false,"given":"Haytham","family":"Rezk","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,1]]},"reference":[{"key":"ref_1","first-page":"150","article-title":"Notes on some points in the integral calculus, (LXIT)","volume":"54","author":"Hardy","year":"1925","journal-title":"Messenger Math."},{"key":"ref_2","first-page":"12","article-title":"Notes on some points in the integral calculus LXIV. 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