New sharp Jordan type inequalities and their applications

2012/06/24 by Zhen-Hang Yang, Yang, Zhen-Hang
Mathematics · #26D05 #26D15 (Primary) 33F05 #26E60 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:26D05 #msc:26D15 #msc:26E60 #msc:33F05

paper · pdf · doi:10.48550/arxiv.1206.5502

10 pages. arXiv admin note: text overlap with arXiv:1206.4911

arxiv created 2012/06/24 · arxiv updated 2012/06/26

Abstract

In this paper, we prove that for x∈(0,π/2) (cos p₀x)^1/p₀<((sin x)/x)<(cos(x/3))³ with the best constants p₀=0.347307245464... and 1/3. Moreover, if p∈ (0,1/3] then the double inequality β_p(cos px)^1/p<((sin x)/x)<(cos px)^1/p holds for x∈(0,π/2), where β_p=2π^-1(cos((pπ)/2))^-1/p and 1 are the best possible. Its reverse one holds if p∈[1/2,1]. As applications, some new inequalities are established.

Citations

• Refinements of Mitrinović-Cusa inequality

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