Sharp Extensions of a Cusa-Huygens Type Inequality
DOI:
https://doi.org/10.52737/18291163-2024.16.14-1-12Keywords:
Cusa-Huygens Type Inequality, Sharp Bounds, l'Hospital's Rule of MonotonicityAbstract
In this article, we propose an extension and generalization of a Cusa-Huygens type inequality and thus refine an existing inequality in the literature. As an application, we extend the improved Shafer's inequality for the arctangent function.
References
Y.J. Bagul, B. Banjac, C. Chesneau, M. Kostić and B. Malešević, New refinements of Cusa-Huygens inequality. Results Math., 76 (2021), Article no. 107.
Y.J. Bagul and C. Chesneau, Refined forms of Oppenheim and Cusa-Huygens type inequalities. Acta Comment. Univ. Tartu. Math., 24 (2020), no. 2, pp. 183-194.
Y.J. Bagul, C. Chesneau and M. Kostić, On the Cusa-Huygens inequality. RACSAM., 15 (2021), Article no. 29.
Y.J. Bagul, C. Chesneau and M. Kostić, The Cusa-Huygens inequality revisited. Novi Sad J. Math., 50 (2020), no. 2, pp. 149-159.
G. Bercu, Fourier series method related to Wilker-Cusa-Huygens inequalities. Math. Inequal. Appl., 22 (2019), no. 4, pp. 1091-1098.
B. Chaouchi, V.E. Fedorov and M. Kostić, Monotonicity of certain classes of functions related with Cusa-Huygens inequality. Chelyab. Fiz.-Mat. Zh., 6 (2021), no. 3, pp. 331-337.
C.-P. Chen and W.-S. Cheung, Sharp Cusa and Becker-Stark inequalities. J. Inequal. Appl., 2011 (2011), Article No. 136.
C.-P. Chen and J. Sándor, Inequality chains for Wilker, Huygens and Lazarević type inequalities. J. Math. Inequal., 8 (2014), no. 1, pp. 55-67.
R.M. Dhaigude, C. Chesneau and Y.J. Bagul, About trigonometric-polynomial bounds of sinc function. Math. Sci. Appl. E-Notes, 8 (2020), no. 1, pp. 100-104.
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Elsevier, 2007.
C. Huygens, Oeuvres Completes, Société Hollandaise des Sciences, Haga, 1888-1940.
B. Malešević, M. Nenezić, L. Zhu, B. Banjac and M. Petrović, Some new estimates of precision of Cusa-Huygens and Huygens approximations. Appl. Anal. Discrete Math., 15 (2021), no. 1, pp. 243-259.
B. Malešević, T. Lutovac, M. Rašajski and C. Mortici, Extensions of the natural approach to refinements and generalizations of some trigonometric inequalities. Adv. Difference Equ., 2018 (2018), Article no. 90.
B. Malešević and M. Makragić, A method for proving some inequalities on mixed trigonometric polynomial functions. J. Math. Inequal., 10 (2016), no. 3, pp. 849-876.
D.S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin, 1970.
C. Mortici, The natural approach of Wilker-Cusa-Huygens Inequalities. Math. Inequal. Appl., 14 (2011), no. 3, pp. 535-541.
E. Neuman and J. Sándor, On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker and Huygens inequalities. Math. Inequal. Appl., 13 (2010), no. 4, pp. 715-723.
J. Sándor, Sharp Cusa-Huygens and related inequalities. Notes on Number Theory and Discrete Mathematics, 19 (2013), no. 1, pp. 50-54.
J. Sándor and R. Oláh-Gal, On Cusa-Huygens type trigonometric and hyperbolic inequalities. Acta. Univ. Sapientiae Mathematica, 4 (2012), no. 2, pp. 145-153.
R.E. Shafer, Problem E 1867. Amer. Math. Monthly, 73(1966), p. 309.
Z.-H. Yang, Refinements of a two-sided inequality for trigonometric functions. J. Math. Inequal., 7 (2013), no. 4, pp. 601-615.
Z.-H. Yang and Y.-M. Chu, A note on Jordan, Adamović-Mitrinović, and Cusa inequalities. Abst. Appl. Anal., 2014 (2014), Article ID: 364076, 12 pp.
Z.-H. Yang, Y.-M. Chu, Y.-Q. Song and Y.-M. Li, A sharp double inequality for trigonometric functions and its applications. Abst. Appl. Anal., 2014 (2014), Article ID: 592085, 9 pp.
Z.-H. Yang, Y.-L. Jiang, Y.-Q. Song and Y.-M. Chu, Sharp inequalities for trigonometric functions. Abst. Appl. Anal., 2014 (2014), Article ID: 601839, 18 pp.
Z.-H. Yang and Y.-M. Chu, A sharp double inequality involving trigonometric functions and its applications. J. Math. Inequal., 10 (2016), no. 2, pp. 423-432.
L. Zhu, New inequalities of Cusa-Huygens type. Mathematics, 9 (2021), no. 17, 2101.
L. Zhu, New bounds for the sine function and tangent function. Mathematics, 9 (2021), no. 19, 2373.
L. Zhu, New Cusa-Huygens type inequalities. AIMS Mathematics, 5 (2020), no. 5, pp. 5320-5331.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Armenian Journal of Mathematics
This work is licensed under a Creative Commons Attribution 4.0 International License.