Optimal evaluations for the Sándor-Yang mean by power mean

2015/06/25 by Zhen-Hang Yang, Yu‐Ming Chu, Yang, Zhen-Hang +2
Mathematics · #26E60 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Inequalities and Applications #Numerical methods in inverse problems #Point processes and geometric inequalities #Spectral Theory in Mathematical Physics #math.CA #msc:26E60

paper · pdf · doi:10.48550/arxiv.1506.07777

9 pages

arxiv created 2015/06/25 · openalex publication_date 2015/06/25 · arxiv updated 2015/06/26 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that the double inequality M_p(a,b)<B(a,b)<M_q(a,b)% holds for all a, b>0 with a≠ b if and only if p≤ 4log 2/(4+2log 2-π)=1.2351⋯ and q≥ 4/3, where % M_r(a,b)=[(a^r+b^r)/2]^1/r (r≠ 0) and M₀(a,b)=√(ab) is the rth power mean, B(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1} is the Sá% ndor-Yang mean, A(a,b)=(a+b)/2, Q(a,b)=√(a²+b²)/2 and % T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))].

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