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

H. Alzer; H. L. Pedersen

doi:10.1007/s10476-025-00069-6

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

H. Alzer^*, H. L. Pedersen

^*Corresponding author af dette arbejde

Institut for Matematiske Fag

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

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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 λn and μn such that the inequalities (Formula presented.) and (Formula presented.) hold for all x,y∈(0,π) with x+y≤π.

Originalsprog	Engelsk
Tidsskrift	Analysis Mathematica
Vol/bind	51
Udgave nummer	1
Sider (fra-til)	63-73
Antal sider	11
ISSN	0133-3852
DOI	https://doi.org/10.1007/s10476-025-00069-6
Status	Udgivet - 2025

Bibliografisk note

Publisher Copyright:
© The Author(s), under exclusive licence to Akadémiai Kiadó Zrt 2025.

Adgang til dokumentet

10.1007/s10476-025-00069-6

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Citationsformater

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AB - 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 λn and μn such that the inequalities (Formula presented.) and (Formula presented.) hold for all x,y∈(0,π) with x+y≤π.

KW - Bernoulli number

KW - completely and absolutely monotonic

KW - Euler number

KW - Fejér’s sine polynomial

KW - inequality

KW - partialfraction decomposition

KW - sub- and superadditive

KW - trigonometric function

KW - π

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JO - Analysis Mathematica

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