{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:14:06Z","timestamp":1753881246704,"version":"3.41.2"},"reference-count":32,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,2,22]],"date-time":"2012-02-22T00:00:00Z","timestamp":1329868800000},"content-version":"vor","delay-in-days":52,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100009022","name":"Zhejiang Broadcast and TV University","doi-asserted-by":"crossref","award":["XKT-09G21"],"award-info":[{"award-number":["XKT-09G21"]}],"id":[{"id":"10.13039\/501100009022","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We prove that <jats:italic>\u03b1<\/jats:italic><jats:italic>H<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>)+(1 \u2212 <jats:italic>\u03b1<\/jats:italic>)<jats:italic>L<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) &gt; <jats:italic>M<\/jats:italic><jats:sub>(1\u22124<jats:italic>\u03b1<\/jats:italic>)\/3<\/jats:sub>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) for <jats:italic>\u03b1<\/jats:italic> \u2208 (0, 1) and all <jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic> &gt; 0 with <jats:italic>a<\/jats:italic> \u2260 <jats:italic>b<\/jats:italic> if and only if <jats:italic>\u03b1<\/jats:italic> \u2208 [1\/4, 1) and <jats:italic>\u03b1<\/jats:italic><jats:italic>H<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>)+(1 \u2212 <jats:italic>\u03b1<\/jats:italic>)<jats:italic>L<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) &lt; <jats:italic>M<\/jats:italic><jats:sub>(1\u22124<jats:italic>\u03b1<\/jats:italic>)\/3<\/jats:sub>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) if and only if , and the parameter (1 \u2212 4<jats:italic>\u03b1<\/jats:italic>)\/3 is the best possible in either case. Here, <jats:italic>H<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) = 2<jats:italic>a<\/jats:italic><jats:italic>b<\/jats:italic>\/(<jats:italic>a<\/jats:italic> + <jats:italic>b<\/jats:italic>),  <jats:italic>L<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) = (<jats:italic>a<\/jats:italic> \u2212 <jats:italic>b<\/jats:italic>)\/(log<jats:italic>\u2009a<\/jats:italic> \u2212 log<jats:italic>\u2009b<\/jats:italic>), and  <jats:italic>M<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) = ((<jats:italic>a<\/jats:italic><jats:sup><jats:italic>p<\/jats:italic><\/jats:sup> + <jats:italic>b<\/jats:italic><jats:sup><jats:italic>p<\/jats:italic><\/jats:sup>)\/2)\u2009<jats:sup>1\/<jats:italic>p<\/jats:italic><\/jats:sup> (<jats:italic>p<\/jats:italic> \u2260 0)\n and  are the harmonic, logarithmic, and <jats:italic>p<\/jats:italic>th power means of <jats:italic>a<\/jats:italic> and <jats:italic>b<\/jats:italic>, respectively.<\/jats:p>","DOI":"10.1155\/2012\/471096","type":"journal-article","created":{"date-parts":[[2012,2,22]],"date-time":"2012-02-22T21:02:21Z","timestamp":1329944541000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means"],"prefix":"10.1155","volume":"2012","author":[{"given":"Wei-Mao","family":"Qian","sequence":"first","affiliation":[]},{"given":"Zhong-Hua","family":"Shen","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,2,22]]},"reference":[{"key":"e_1_2_5_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.04.032"},{"key":"e_1_2_5_2_2","doi-asserted-by":"publisher","DOI":"10.7153\/jmi-05-27"},{"key":"e_1_2_5_3_2","doi-asserted-by":"publisher","DOI":"10.1155\/2010\/303286"},{"key":"e_1_2_5_4_2","doi-asserted-by":"publisher","DOI":"10.2307\/2317088"},{"key":"e_1_2_5_5_2","doi-asserted-by":"publisher","DOI":"10.2307\/2319447"},{"key":"e_1_2_5_6_2","first-page":"15","article-title":"Inequalities between arithmetic and logarithmic means","author":"Pittenger A. 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