{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T11:35:35Z","timestamp":1771673735560,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,18]],"date-time":"2020-10-18T00:00:00Z","timestamp":1602979200000},"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 investigate some new generalizations and refinements for H\u00f6lder\u2019s inequality and it\u2019s reverse on time scales through the diamond-\u03b1 dynamic integral, which is defined as a linear combination of the delta and nabla integrals, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. Our results as special cases extend some integral dynamic inequalities and Qi\u2019s inequalities achieved on time scales and also include some integral disparities as particular cases when T=R.<\/jats:p>","DOI":"10.3390\/sym12101716","type":"journal-article","created":{"date-parts":[[2020,10,18]],"date-time":"2020-10-18T21:26:06Z","timestamp":1603056366000},"page":"1716","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["More on H\u00f6lder\u2019s Inequality and It\u2019s Reverse via the Diamond-Alpha Integral"],"prefix":"10.3390","volume":"12","author":[{"given":"M.","family":"Zakarya","sequence":"first","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":"H. A.","family":"Abd El-Hamid","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Beni- Suef University, Beni-Suef, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2222-7973","authenticated-orcid":false,"given":"Ghada","family":"AlNemer","sequence":"additional","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\/0000-0001-6782-7908","authenticated-orcid":false,"given":"H. M.","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":[[2020,10,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1155\/S0161171297000264","article-title":"On an inverse to the H\u00f6lder inequality","volume":"20","author":"Pecaric","year":"1997","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1016\/j.camwa.2010.11.015","article-title":"Generalizations of H\u00f6lder\u2019s and some related inequalities","volume":"61","author":"Qiang","year":"2011","journal-title":"Compt. Math. 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