{"title":"函数的高斯奇调和平均不等式的改进","authors":"Horst Alzer","doi":"10.4171/rsmup/152","DOIUrl":null,"url":null,"abstract":"In 1974, W. Gautschi proved that $$ 1<\\frac{2}{1/\\Gamma(x) +1/\\Gamma(1/x)} \\quad \\textrm{for} \\quad 0\\<x\\neq 1. $$ Here, we present the following refinement: $$ 1<\\Gamma\\Bigl( \\frac{2}{x+1/x}\\Bigr)< \\frac{2}{1/\\Gamma(x) +1/\\Gamma(1/x)}, \\quad 0\\<x\\neq 1. $$","PeriodicalId":20997,"journal":{"name":"Rendiconti del Seminario Matematico della Università di Padova","volume":"8 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Refinement of Gautschi's harmonic mean inequality for the gamma function\",\"authors\":\"Horst Alzer\",\"doi\":\"10.4171/rsmup/152\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1974, W. Gautschi proved that $$ 1<\\\\frac{2}{1/\\\\Gamma(x) +1/\\\\Gamma(1/x)} \\\\quad \\\\textrm{for} \\\\quad 0\\\\<x\\\\neq 1. $$ Here, we present the following refinement: $$ 1<\\\\Gamma\\\\Bigl( \\\\frac{2}{x+1/x}\\\\Bigr)< \\\\frac{2}{1/\\\\Gamma(x) +1/\\\\Gamma(1/x)}, \\\\quad 0\\\\<x\\\\neq 1. $$\",\"PeriodicalId\":20997,\"journal\":{\"name\":\"Rendiconti del Seminario Matematico della Università di Padova\",\"volume\":\"8 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-10-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Rendiconti del Seminario Matematico della Università di Padova\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4171/rsmup/152\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Rendiconti del Seminario Matematico della Università di Padova","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/rsmup/152","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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