Sharp Inequalities between Harmonic, Seiffert, Quadratic and Contraharmonic Means

2012/10/15 by Gen-Di Wang, Wang, Gen-Di, Chen-Yan Yang +3
Mathematics · #26E60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #G.1.6 #acm:26E60 #math.CA #msc:26E60

paper · pdf · doi:10.48550/arxiv.1210.3875

11 pages

arxiv created 2012/10/15 · arxiv updated 2012/10/16

Abstract

In this paper, we present the greatest values α, λ and p, and the least values β, μ and q such that the double inequalities αD(a,b)+(1-α)H(a,b)<T(a,b)<βD(a,b)+(1-β) H(a,b), λD(a,b)+(1-λ)H(a,b)<C(a,b)<μD(a,b)+(1-μ) H(a,b) and p D(a,b)+(1-p)H(a,b)<Q(a,b)<q D(a,b)+(1-q)H(a,b) hold for all a,b>0 with a≠ b, where H(a,b)=2ab/(a+b), T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))], Q(a,b)=√((a²+b²)/2), C(a,b)=(a²+b²)/(a+b) and D(a,b)=(a³+b³)/(a²+b²) are the harmonic, Seiffert, quadratic, first contraharmonic and second contraharmonic means of a and b, respectively.

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