New inequalities of Huygens-type involving tangent and sine functions

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.

Keywords

• circular functions
• refinements of the Huygens-type inequalities
• even-indexed Bernoulli number
• Euler number

Supporting Institution

the National Natural Science Foundation of China (no. 61772025). The second author was supported in part by the Serbian Ministry of Education, Science and Technological Development, under projects ON 174032 and III 44006.

Project Number

the National Natural Science Foundation of China (no. 61772025). The second author was supported in part by the Serbian Ministry of Education, Science and Technological Development, under projects ON 174032 and III 44006.

Thanks

The authors are grateful to anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

References

1. [1] C. Huygens, Oeuvres completes, publiees par la Societe hollandaise des science, Haga, 1888–1940 (20 volumes).
2. [2] F.T. Campan, The Story of Number, in: Ed. Albatros (Ed), Romania, 1977.
3. [3] E. Neuman, On Wilker and Huygens type inequalities, Math. Inequal. Appl. 15 (2), 271–279, 2012.
4. [4] Ch.-P. Chen and W.-S. Cheung, Sharpness of Wilker and Huygens Type Inequalities, J. Inequal. Appl. 2012 (1), 1–11, 2012.
5. [5] J.-L. Li, An identity related to Jordan’s inequality, Int. J. Math. Math. Sci. 2006, Art. id 76782, 2006.
6. [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. [7] A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Elsevier Academic Press, San Diego, Calif, USA, 3rd edition, 2004.
8. [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. [9] H. Alzer, Sharp bounds for the Bernoulli numbers, Archiv der Mathematik, 74 (3), 207–211, 2000.