{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T19:52:57Z","timestamp":1760385177505,"version":"3.41.2"},"reference-count":10,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,11,23]],"date-time":"2011-11-23T00:00:00Z","timestamp":1322006400000},"content-version":"vor","delay-in-days":326,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11071069","T200924"],"award-info":[{"award-number":["11071069","T200924"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008867","name":"Department of Education of Zhejiang Province","doi-asserted-by":"crossref","award":["11071069","T200924"],"award-info":[{"award-number":["11071069","T200924"]}],"id":[{"id":"10.13039\/501100008867","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":[[2011,1]]},"abstract":"<jats:p>For <jats:italic>\u03b1<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic> \u2208 (0, 1\/2) we prove that the double inequality <jats:italic>G<\/jats:italic>(<jats:italic>\u03b1<\/jats:italic><jats:italic>a<\/jats:italic> + (1 \u2212 <jats:italic>\u03b1<\/jats:italic>)<jats:italic>b<\/jats:italic>, <jats:italic>\u03b1<\/jats:italic><jats:italic>b<\/jats:italic> + (1 \u2212 <jats:italic>\u03b1<\/jats:italic>)<jats:italic>a<\/jats:italic>)&lt;<jats:italic>P<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) &lt; <jats:italic>G<\/jats:italic>(<jats:italic>\u03b2<\/jats:italic><jats:italic>a<\/jats:italic> + (1 \u2212 <jats:italic>\u03b2<\/jats:italic>)<jats:italic>b<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic><jats:italic>b<\/jats:italic> + (1 \u2212 <jats:italic>\u03b2<\/jats:italic>)<jats:italic>a<\/jats:italic>) holds for 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  and . Here, <jats:italic>G<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) and <jats:italic>P<\/jats:italic>(<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) denote the geometric and Seiffert means of two positive numbers <jats:italic>a<\/jats:italic> and <jats:italic>b<\/jats:italic>, respectively.<\/jats:p>","DOI":"10.1155\/2011\/261237","type":"journal-article","created":{"date-parts":[[2011,11,23]],"date-time":"2011-11-23T21:06:32Z","timestamp":1322082392000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Optimal Double Inequality between Seiffert and Geometric Means"],"prefix":"10.1155","volume":"2011","author":[{"given":"Yu-Ming","family":"Chu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Miao-Kun","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zi-Kui","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2011,11,23]]},"reference":[{"key":"e_1_2_4_1_2","article-title":"Problem 887","volume":"11","author":"Seiffert H.-J.","year":"1993","journal-title":"Nieuw Archief voor Wiskunde (Series 4)"},{"key":"e_1_2_4_2_2","doi-asserted-by":"publisher","DOI":"10.1155\/2010\/436457"},{"key":"e_1_2_4_3_2","doi-asserted-by":"crossref","first-page":"47","DOI":"10.7153\/mia-07-06","article-title":"Optimal inequalities between Seiffert\u2032s mean and power means","volume":"7","author":"H\u00e4st\u00f6 P. 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A.","year":"1994","journal-title":"Nieuw Archief voor Wiskunde (Series 4)"},{"key":"e_1_2_4_5_2","doi-asserted-by":"publisher","DOI":"10.1155\/2011\/686834"},{"key":"e_1_2_4_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/s000130050539"},{"key":"e_1_2_4_7_2","first-page":"195","article-title":"Ungleichungen f\u00fcr einen bestimmten Mittelwert","volume":"13","author":"Seiffert H.-J.","year":"1995","journal-title":"Nieuw Archief voor Wiskunde (Series 4)"},{"key":"e_1_2_4_8_2","first-page":"1079","article-title":"The best bounds of the combination of arithmetic and harmonic means for the Seiffert\u2032s mean","volume":"4","author":"Wang S.-S.","year":"2010","journal-title":"International Journal of Mathematical Analysis"},{"key":"e_1_2_4_9_2","doi-asserted-by":"crossref","first-page":"581","DOI":"10.7153\/jmi-04-51","article-title":"Sharp bounds for Seiffert means in terms of Lehmer means","volume":"4","author":"Wang M.-K.","year":"2010","journal-title":"Journal of Mathematical Inequalities"},{"key":"e_1_2_4_10_2","doi-asserted-by":"publisher","DOI":"10.1155\/2010\/108920"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2011\/261237.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2011\/261237.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2011\/261237","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,10]],"date-time":"2024-06-10T15:06:33Z","timestamp":1718031993000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2011\/261237"}},"subtitle":[],"editor":[{"given":"J. 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