Inequalities for 1/(1-cos(x)) and its derivatives

2024/06/13 by Horst Alzer, Alzer, Horst, H. Pedersen +1 · 1 citation
Mathematics · #Mathematical Inequalities and Applications #Mathematics and Applications #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2406.08932

Abstract

We prove that the function g(x)= 1 / ( 1 - cos(x) ) is completely monotonic on (0,π] and absolutely monotonic on [π, 2π), and we determine the best possible bounds λ_n and μ_n such that the inequalities λ_n ≤ g⁽ⁿ⁾(x)+g⁽ⁿ⁾(y)-g⁽ⁿ⁾(x+y) (n ≥ 0 even) and μ_n ≤ g⁽ⁿ⁾(x+y)-g⁽ⁿ⁾(x)-g⁽ⁿ⁾(y) (n ≥ 1 odd) hold for all x,y∈ (0,π) with x+y≤ π.

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