{"title":"New inequalities for operator monotone functions","authors":"S. Dragomir","doi":"10.52846/ami.v48i1.1410","DOIUrl":null,"url":null,"abstract":"\"In this paper we prove that, if f:[0,∞)→R is operator monotone on [0,∞), then for all A, B such that 0<α≤A≤β<γ≤B≤δ for some positive constants α, β, γ, δ, 0≤(γ-β)((f(δ)-f(β))/(δ-β))≤f(B)-f(A)≤(δ-α)((f(γ)-f(α))/(γ-α)). In particular, we have the refinement and reverse of the celebrated Löwner-Heinz inequality 0<(γ-β)((δ^{r}-β^{r})/(δ-β))≤B^{r}-A^{r}≤(δ-α)((γ^{r}-α^{r})/(γ-α)) for all r∈(0,1].\"","PeriodicalId":43654,"journal":{"name":"Annals of the University of Craiova-Mathematics and Computer Science Series","volume":"36 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of the University of Craiova-Mathematics and Computer Science Series","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.52846/ami.v48i1.1410","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
"In this paper we prove that, if f:[0,∞)→R is operator monotone on [0,∞), then for all A, B such that 0<α≤A≤β<γ≤B≤δ for some positive constants α, β, γ, δ, 0≤(γ-β)((f(δ)-f(β))/(δ-β))≤f(B)-f(A)≤(δ-α)((f(γ)-f(α))/(γ-α)). In particular, we have the refinement and reverse of the celebrated Löwner-Heinz inequality 0<(γ-β)((δ^{r}-β^{r})/(δ-β))≤B^{r}-A^{r}≤(δ-α)((γ^{r}-α^{r})/(γ-α)) for all r∈(0,1]."