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
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Authors and affiliations

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

Abstract

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.

Keywords

  • Inequalities
  • Trigonometric functions
  • Hyperbolic functions
  • Monotonicity theorems

AMS Classification

  • 26D05
  • 26D07
  • 26D99

References

  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

APA

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

Chicago

Sándor, József. “Sharp Cusa–Huygens and related inequalities.” Notes on Number Theory and Discrete Mathematics 19, no. 1 (2013): 50-54.

MLA

Sándor, József. “Sharp Cusa–Huygens and related inequalities.” Notes on Number Theory and Discrete Mathematics 19.1 (2013): 50-54. Print.

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