Sharp two parameter bounds for logarithmic and arithmetic-geometric means

2012/09/15 by Yu‐Ming Chu, Yu-Ming Chu, Ye-Fang Qiu +6
Mathematics · #26E60 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #G.1.2 #Mathematical Inequalities and Applications #acm:26E60 #math.CA #msc:26E60

paper · pdf · doi:10.48550/arxiv.1209.3350

8 pages

arxiv created 2012/09/15 · openalex publication_date 2012/09/15 · arxiv updated 2012/11/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For fixed s≥ 1 and t₁,t₂∈(0,1/2) we prove that the inequalities G^s(t₁a+(1-t₁)b,t₁b+(1-t₁)a)A^1-s(a,b)>AG(a,b) and G^s(t₂a+(1-t₂)b,t₂b+(1-t₂)a)A^1-s(a,b)>L(a,b) hold for all a,b>0 with a≠ b if and only if t₁≥ 1/2-√(2s)/(4s) and t₂≥ 1/2-√(6s)/(6s). Here G(a,b), L(a,b), AG(a,b) and A(a,b) are the geometric, logarithmic, arithmetic-geometric and arithmetic means of a and b, respectively.

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