{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T04:16:36Z","timestamp":1766376996431,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,9,4]],"date-time":"2018-09-04T00:00:00Z","timestamp":1536019200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11501141"],"award-info":[{"award-number":["11501141"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Top-notch Talents Support  Project of Education Department of Guizhou Province","award":["QJHKYZ [2016]066"],"award-info":[{"award-number":["QJHKYZ [2016]066"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the current note, we investigate the mathematical relations among the weighted arithmetic mean\u2013geometric mean (AM\u2013GM) inequality, the H\u00f6lder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM\u2013GM inequality, the weighted power-mean inequality and the H\u00f6lder inequality are fully achieved. The new results are more generalized than those of previous studies.<\/jats:p>","DOI":"10.3390\/sym10090380","type":"journal-article","created":{"date-parts":[[2018,9,5]],"date-time":"2018-09-05T03:08:55Z","timestamp":1536116935000},"page":"380","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["The Weighted Arithmetic Mean\u2013Geometric Mean Inequality is Equivalent to the H\u00f6lder Inequality"],"prefix":"10.3390","volume":"10","author":[{"given":"Yongtao","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7895-2050","authenticated-orcid":false,"given":"Xian-Ming","family":"Gu","sequence":"additional","affiliation":[{"name":"School of Economic Mathematics\/Institute of Mathematics, Southwestern University of Finance and Economic, Chengdu, Sichuan 611130, China"},{"name":"Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianxing","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cvetkovski, Z. (2012). Inequalities: Theorems, Techniques and Selected Problems, Springer.","DOI":"10.1007\/978-3-642-23792-8"},{"key":"ref_2","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. (1934). Inequalities, Cambridge University Press."},{"key":"ref_3","first-page":"1","article-title":"A Survey on Cauchy-Bunyakovsky-Schwarz Type Discrete Inequalities","volume":"4","author":"Dragomir","year":"2003","journal-title":"J. Inequal. Pure Appl. 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[2nd ed.].","DOI":"10.1007\/978-0-387-68276-1"},{"key":"ref_13","unstructured":"Sitnik, S.M. (arXiv, 2010). Generalized Young and Cauchy\u2013Bunyakowsky Inequalities with Applications: A survey, arXiv, (In Russian)."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1007\/s00283-012-9292-1","article-title":"The AM-GM inequality and CBS inequality are equivalent","volume":"34","author":"Lin","year":"2012","journal-title":"Math. Intell."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/9\/380\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:18:45Z","timestamp":1760195925000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/9\/380"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,4]]},"references-count":14,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2018,9]]}},"alternative-id":["sym10090380"],"URL":"https:\/\/doi.org\/10.3390\/sym10090380","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,9,4]]}}}