Refinements of the inequalities between Neuman-Sandor, arithmetic, contra-harmonic and quadratic means

2012/09/13 by Yu‐Ming Chu, Yu-Ming Chu, Chu, Yu-Ming +2
Mathematics · #26E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #G.1.6 #Mathematical Inequalities and Applications #Optimization and Control (math.OC) #acm:26E20 #math.CA #math.OC #msc:26E20

paper · pdf · doi:10.48550/arxiv.1209.2920

9 pages

arxiv created 2012/09/13 · openalex publication_date 2012/09/13 · arxiv updated 2012/11/03 · openalex created_date 2016/09/16 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that the inequalities α[1/3 Q(a,b)+2/3 A(a,b)]+(1-α)Q^1/3(a,b)A^2/3(a,b)0 with a≠ b if and only if α≤ (3-3√[6]2log(1+√(2)))/[(2+√(2)-3√[6]2)log(1+√(2))]=0.777..., β≥ 4/5, λ≤ (6-6√[6]2log(1+√(2)))/(7-6√[6]2log(1+√(2)))=0.274..., and μ≥ 8/25. Here, M(a,b), A(a,b), C(a,b), and Q(a,b) denote the Neuman-Sándor, arithmetic, contra-harmonic, and quadratic means of a and b, respectively.

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