New sharp Cusa--Huygens type inequalities for trigonometric and hyperbolic functions

2014/08/10 by Zhen-Hang Yang, Yang, Zhen-Hang
Mathematics · #26D05 #26D15 #26E60 #33B10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Inequalities and Applications #Mathematical functions and polynomials #math.CA #msc:26D05 #msc:26D15 #msc:26E60 #msc:33B10

paper · pdf · doi:10.48550/arxiv.1408.2243

15 pages

arxiv created 2014/08/10 · openalex publication_date 2014/08/10 · arxiv updated 2014/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for p∈ (0,1], the double inequality% \tfrac13p²cos px+1-\tfrac13p²<(sin x)/(x)<\tfrac1% 3q²cos qx+1-\tfrac13q²% holds for x∈ (0,π/2) if and only if 0<p≤ p₀≈ 0.77086 and √(15)/5=p₁≤ q≤ 1. While its hyperbolic version holds for % x>0 if and only if 0<p≤ p₁=√(15)/5 and q≥ 1. As applications, some more accurate estimates for certain mathematical constants are derived, and some new and sharp inequalities for Schwab-Borchardt mean and logarithmic means are established.

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