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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 52 Issue: 1

New inequalities of Huygens-type involving tangent and sine functions

Ling Zhu(Zhejiang Gongshang University)0000-0002-9059-3983
Branko Malesevic(University of Belgrade)0000-0002-4963-4149
Pages
—
Publication Date
2023-02-15
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⚡ TL;DR(Too Long; Didn't Read)

Using the estimations of the even-indexed Bernoulli number and Euler number this paper established some new inequalities for the three functions $2\left( \sin x\right) /x+\left( \tan x\right) /x$, $\left( \sin x\right) /x+2\left( \tan (x/2)\right) /\left( x/2\right) $ and $2x/\sin x+x/\tan x$ bounded by the powers of tangent function.

📌 3 Key Takeaways

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Objective & Hypothesis:Using the estimations of the even-indexed Bernoulli number and Euler number this paper established some new inequalities for the three functions $2\left( \sin x\right) /x+\left( \tan x\right) /x$, $\left( \sin x\right) /x+2\left( \tan (x/2)\right) /\left( x/2\right) $ and $2x/\sin x+x/\tan x$ bounded by the powers of tangent function.
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Key Findings & Data:Using the estimations of the even-indexed Bernoulli number and Euler number this paper established some new inequalities for the three functions $2\left( \sin x\right) /x+\left( \tan x\right) /x$, $\left( \sin x\right) /x+2\left( \tan (x/2)\right) /\left( x/2\right) $ and $2x/\sin x+x/\tan x$ bounded by the powers of tangent function.
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Conclusion & Impact:Using the estimations of the even-indexed Bernoulli number and Euler number this paper established some new inequalities for the three functions $2\left( \sin x\right) /x+\left( \tan x\right) /x$, $\left( \sin x\right) /x+2\left( \tan (x/2)\right) /\left( x/2\right) $ and $2x/\sin x+x/\tan x$ bounded by the powers of tangent function.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:circular functions · refinements of the Huygens-type inequalities · even-indexed Bernoulli number

Abstract

Using the estimations of the even-indexed Bernoulli number and Euler number this paper established some new inequalities for the three functions $2\left( \sin x\right) /x+\left( \tan x\right) /x$, $\left( \sin x\right) /x+2\left( \tan (x/2)\right) /\left( x/2\right) $ and $2x/\sin x+x/\tan x$ bounded by the powers of tangent function.

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Cite This Article

Zhu, L., Malesevic, B. (2023). New inequalities of Huygens-type involving tangent and sine functions. Hacettepe Journal of Mathematics and Statistics, Volume: 52 Issue: 1. https://doi.org/10.15672/hujms.951700

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References (9)

  1. C. Huygens, Oeuvres completes, publiees par la Societe hollandaise des science, Haga, 1888–1940 (20 volumes).
  2. F.T. Campan, The Story of Number, in: Ed. Albatros (Ed), Romania, 1977.
  3. E. Neuman, On Wilker and Huygens type inequalities, Math. Inequal. Appl. 15 (2), 271–279, 2012.
  4. Ch.-P. Chen and W.-S. Cheung, Sharpness of Wilker and Huygens Type Inequalities, J. Inequal. Appl. 2012 (1), 1–11, 2012.
  5. J.-L. Li, An identity related to Jordan’s inequality, Int. J. Math. Math. Sci. 2006, Art. id 76782, 2006.
  6. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. National Bureau of Standards, Washington, DC, USA, 1964.
  7. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Elsevier Academic Press, San Diego, Calif, USA, 3rd edition, 2004.
  8. C. D’Aniello, On some inequalities for the Bernoulli numbers, Rendiconti del Circolo Matematico di Palermo. Serie II 43 (3), 329–332, 1994.
  9. H. Alzer, Sharp bounds for the Bernoulli numbers, Archiv der Mathematik, 74 (3), 207–211, 2000.