Sharp Cusa–Huygens and related inequalities

József Sándor
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 1, Pages 50–54
Full paper (PDF, 130 Kb)


Authors and affiliations

József Sándor
Babeș-Bolyai University, Department of Mathematics
Str. Kogălniceanu nr. 1, 400084 Cluj-Napoca, Romania


We determine the best positive constants a and b such that
\left(\ds\f{\cos x+2}{3}\right)^a<\ds\f{\sin x}{x}<\left(\ds\f{\cos x+2}{3}\right)^b.
Similar sharp inequalities are also considered.


  • Inequalities
  • Trigonometric functions
  • Hyperbolic functions
  • Monotonicity theorems

AMS Classification

  • 26D05
  • 26D07
  • 26D99


  1. Hardy, G.H., J.E. Littlewood, G. Pólya. Inequalities, Cambridge Univ. Press, 1959.
  2. Neuman, E., J. Sándor, On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa–Huygens, Wilker, and Huygens inequalities, Math. Ineq. Appl., Vol. 13, 2010, No. 4, 715–723.
  3. Neuman, E., J. Sándor, Optimal inequalities for hyperbolic and trigonometric functions, Bull. Math. Anal. Appl., Vol. 3, 2011, No. 3, 177–181.
  4. Sándor, J., Two sharp inequalities for trigonometric and hyperbolic functions, ath. Ineq. Appl., to appear

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Cite this paper

Sándor, J. (2013). Sharp Cusa–Huygens and related inequalities. Notes on Number Theory and Discrete Mathematics, 19(1), 50-54.

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