{"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}
引用次数: 0
Abstract
In 1974, W. Gautschi proved that $$ 1<\frac{2}{1/\Gamma(x) +1/\Gamma(1/x)} \quad \textrm{for} \quad 0\